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Chapter 5Resource Masters
Geometry
Reading to Learn MathematicsVocabulary Builder
NAME ______________________________________________ DATE ____________ PERIOD _____
55
© Glencoe/McGraw-Hill vii Glencoe Geometry
Voca
bula
ry B
uild
erThis is an alphabetical list of the key vocabulary terms you will learn in Chapter 5.As you study the chapter, complete each term’s definition or description. Rememberto add the page number where you found the term. Add these pages to yourGeometry Study Notebook to review vocabulary at the end of the chapter.
Vocabulary Term Found on Page Definition/Description/Example
altitude
centroid
circumcenter
SUHR·kuhm·SEN·tuhr
concurrent lines
incenter
indirect proof
(continued on the next page)
⎧ ⎪ ⎨ ⎪ ⎩
© Glencoe/McGraw-Hill viii Glencoe Geometry
Vocabulary Term Found on Page Definition/Description/Example
indirect reasoning
median
orthocenter
OHR·thoh·CEN·tuhr
perpendicular bisector
point of concurrency
proof by contradiction
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
Reading to Learn MathematicsVocabulary Builder (continued)
NAME ______________________________________________ DATE ____________ PERIOD _____
55
Learning to Read MathematicsProof Builder
NAME ______________________________________________ DATE ____________ PERIOD _____
55
© Glencoe/McGraw-Hill ix Glencoe Geometry
Proo
f Bu
ilderThis is a list of key theorems and postulates you will learn in Chapter 5. As you
study the chapter, write each theorem or postulate in your own words. Includeillustrations as appropriate. Remember to include the page number where youfound the theorem or postulate. Add this page to your Geometry Study Notebookso you can review the theorems and postulates at the end of the chapter.
Theorem or Postulate Found on Page Description/Illustration/Abbreviation
Theorem 5.1
Theorem 5.2
Theorem 5.3Circumcenter Theorem
Theorem 5.4
Theorem 5.5
Theorem 5.6Incenter Theorem
Theorem 5.7Centroid Theorem
(continued on the next page)
© Glencoe/McGraw-Hill x Glencoe Geometry
Theorem or Postulate Found on Page Description/Illustration/Abbreviation
Theorem 5.8Exterior Angle Inequality Theorem
Theorem 5.9
Theorem 5.10
Theorem 5.11Triangle Inequality Theorem
Theorem 5.12
Theorem 5.13SAS Inequality/Hinge Theorem
Theorem 5.14SSS Inequality
Learning to Read MathematicsProof Builder (continued)
NAME ______________________________________________ DATE ____________ PERIOD _____
55
Study Guide and InterventionBisectors, Medians, and Altitudes
NAME ______________________________________________ DATE ____________ PERIOD _____
5-15-1
© Glencoe/McGraw-Hill 245 Glencoe Geometry
Less
on
5-1
Perpendicular Bisectors and Angle Bisectors A perpendicular bisector of aside of a triangle is a line, segment, or ray that is perpendicular to the side and passesthrough its midpoint. Another special segment, ray, or line is an angle bisector, whichdivides an angle into two congruent angles.
Two properties of perpendicular bisectors are:(1) a point is on the perpendicular bisector of a segment if and only if it is equidistant from
the endpoints of the segment, and(2) the three perpendicular bisectors of the sides of a triangle meet at a point, called the
circumcenter of the triangle, that is equidistant from the three vertices of the triangle.
Two properties of angle bisectors are:(1) a point is on the angle bisector of an angle if and only if it is equidistant from the sides
of the angle, and(2) the three angle bisectors of a triangle meet at a point, called the incenter of the
triangle, that is equidistant from the three sides of the triangle.
BD!!" is the perpendicularbisector of A!C!. Find x.
BD!!" is the perpendicular bisector of A!C!, soAD ! DC.3x " 8 ! 5x # 6
14 ! 2x7 ! x
3x ! 8
5x " 6B
C
D
A
MR!!" is the angle bisectorof !NMP. Find x if m!1 # 5x ! 8 andm!2 # 8x " 16.
MR!!" is the angle bisector of !NMP, so m!1 ! m!2.5x " 8 ! 8x # 16
24 ! 3x8 ! x
12
N R
PM
Example 1Example 1 Example 2Example 2
ExercisesExercises
Find the value of each variable.
1. 2. 3.
DE#!" is the perpendicular "CDF is equilateral. DF!!" bisects !CDE.bisector of A!C!.
4. For what kinds of triangle(s) can the perpendicular bisector of a side also be an anglebisector of the angle opposite the side?
5. For what kind of triangle do the perpendicular bisectors intersect in a point outside thetriangle?
FE
DC(4x ! 30)$
8x $D
F
C
E10y " 46x $
3x $
8y
CE
DA
B
7x " 96x " 2
© Glencoe/McGraw-Hill 246 Glencoe Geometry
Medians and Altitudes A median is a line segment that connects the vertex of atriangle to the midpoint of the opposite side. The three medians of a triangle intersect at thecentroid of the triangle.
Centroid The centroid of a triangle is located two thirds of the distance from aTheorem vertex to the midpoint of the side opposite the vertex on a median.
AL ! $23$AE, BL ! $
23$BF, CL ! $
23$CD
Points R, S, and T are the midpoints of A!B!, B!C! and A!C!, respectively. Find x, y, and z.
CU ! $23$CR BU ! $
23$BT AU ! $
23$AS
6x ! $23$(6x " 15) 24 ! $
23$(24 " 3y # 3) 6z " 4 ! $
23$(6z " 4 " 11)
9x ! 6x " 15 36 ! 24 " 3y # 3 $32$(6z " 4) ! 6z " 4 " 11
3x ! 15 36 ! 21 " 3y 9z " 6 ! 6z " 15x ! 5 15 ! 3y 3z ! 9
5 ! y z ! 3
Find the value of each variable.
1. 2.
B!D! is a median. AB ! CB; D, E, and F are midpoints.
3. 4.
EH ! FH ! HG
5. 6.
V is the centroid of "RST;D is the centroid of "ABC. TP ! 18; MS ! 15; RN ! 24
7. For what kind of triangle are the medians and angle bisectors the same segments?
8. For what kind of triangle is the centroid outside the triangle?
P
M
V
T
N
R S
y
x
z
G
FE
B
A C
24
329z ! 6 6z
6x
8y
MJ
PN
O
L
K3y ! 5
2x6z
122410
H GF
E
7x ! 4
9x " 2
5y
DB
E
F
A
C
9x ! 6
10x
3y
15D
BA
C
6x ! 3
7x " 1
A CT
SR U
B
3y " 3
6x
1524
11
6z ! 4
A CF
EDL
Bcentroid
Study Guide and Intervention (continued)
Bisectors, Medians, and Altitudes
NAME ______________________________________________ DATE ____________ PERIOD _____
5-15-1
ExampleExample
ExercisesExercises
Skills PracticeBisectors, Medians, and Altitudes
NAME ______________________________________________ DATE ____________ PERIOD _____
5-15-1
© Glencoe/McGraw-Hill 247 Glencoe Geometry
Less
on
5-1
ALGEBRA For Exercises 1–4, use the given information to find each value.
1. Find x if E!G! is a median of "DEF. 2. Find x and RT if S!U! is a median of "RST.
3. Find x and EF if B!D! is an angle bisector. 4. Find x and IJ if H!K! is an altitude of "HIJ.
ALGEBRA For Exercises 5–7, use the following information.In "LMN, P, Q, and R are the midpoints of L!M!, M!N!, and L!N!,respectively.
5. Find x.
6. Find y.
7. Find z.
ALGEBRA Lines a, b, and c are perpendicular bisectors of "PQR and meet at A.
8. Find x.
9. Find y.
10. Find z.
COORDINATE GEOMETRY The vertices of "HIJ are G(1, 0), H(6, 0), and I(3, 6). Findthe coordinates of the points of concurrency of "HIJ.
11. orthocenter 12. centroid 13. circumcenter
5y " 6
8x ! 16
7z ! 4
24
18
R QA
ab c
P
y ! 1
2z2.8
23.6
x
L
NQ
RB
P
M
(3x ! 3)$
x ! 8
x " 9
I
JH
K
A
D4x " 1
2x ! 6B
G
E
F
C
R
U5x " 30
2x ! 24
S
T
D
G3x ! 1
5x " 17E
F
© Glencoe/McGraw-Hill 248 Glencoe Geometry
ALGEBRA In "ABC, B!F! is the angle bisector of !ABC, A!E!, B!F!,and C!D! are medians, and P is the centroid.
1. Find x if DP ! 4x # 3 and CP ! 30.
2. Find y if AP ! y and EP ! 18.
3. Find z if FP ! 5z " 10 and BP ! 42.
4. If m!ABC ! x and m!BAC ! m!BCA ! 2x # 10, is B!F! an altitude? Explain.
ALGEBRA In "PRS, P!T! is an altitude and P!X! is a median.
5. Find RS if RX ! x " 7 and SX ! 3x # 11.
6. Find RT if RT ! x # 6 and m!PTR ! 8x # 6.
ALGEBRA In "DEF, G!I! is a perpendicular bisector.
7. Find x if EH ! 16 and FH ! 6x # 5.
8. Find y if EG ! 3.2y # 1 and FG ! 2y " 5.
9. Find z if m!EGH ! 12z.
COORDINATE GEOMETRY The vertices of "STU are S(0, 1), T(4, 7), and U(8, "3).Find the coordinates of the points of concurrency of "STU.
10. orthocenter 11. centroid 12. circumcenter
13. MOBILES Nabuko wants to construct a mobile out of flat triangles so that the surfacesof the triangles hang parallel to the floor when the mobile is suspended. How canNabuko be certain that she hangs the triangles to achieve this effect?
D I
HF
G
E
S R
P
TX
A
C
F
E
DP
B
Practice Bisectors, Medians, and Altitudes
NAME ______________________________________________ DATE ____________ PERIOD _____
5-15-1
Reading to Learn MathematicsBisectors, Medians, and Altitudes
NAME ______________________________________________ DATE ____________ PERIOD _____
5-15-1
© Glencoe/McGraw-Hill 249 Glencoe Geometry
Less
on
5-1
Pre-Activity How can you balance a paper triangle on a pencil point?
Read the introduction to Lesson 5-1 at the top of page 238 in your textbook.
Draw any triangle and connect each vertex to the midpoint of the oppositeside to form the three medians of the triangle. Is the point where the threemedians intersect the midpoint of each of the medians?
Reading the Lesson
1. Underline the correct word or phrase to complete each sentence.
a. Three or more lines that intersect at a common point are called(parallel/perpendicular/concurrent) lines.
b. Any point on the perpendicular bisector of a segment is (parallel to/congruent to/equidistant from) the endpoints of the segment.
c. A(n) (altitude/angle bisector/median/perpendicular bisector) of a triangle is a segment drawn from a vertex of the triangle perpendicular to the line containing the opposite side.
d. The point of concurrency of the three perpendicular bisectors of a triangle is called the(orthocenter/circumcenter/centroid/incenter).
e. Any point in the interior of an angle that is equidistant from the sides of that angle lies on the (median/angle bisector/altitude).
f. The point of concurrency of the three angle bisectors of a triangle is called the(orthocenter/circumcenter/centroid/incenter).
2. In the figure, E is the midpoint of A!B!, F is the midpoint of B!C!,and G is the midpoint of A!C!.
a. Name the altitudes of "ABC.b. Name the medians of "ABC.c. Name the centroid of "ABC.d. Name the orthocenter of "ABC.e. If AF ! 12 and CE ! 9, find AH and HE.
Helping You Remember
3. A good way to remember something is to explain it to someone else. Suppose that aclassmate is having trouble remembering whether the center of gravity of a triangle isthe orthocenter, the centroid, the incenter, or the circumcenter of the triangle. Suggest away to remember which point it is.
A B
C
FG
E D
H
© Glencoe/McGraw-Hill 250 Glencoe Geometry
Inscribed and Circumscribed CirclesThe three angle bisectors of a triangle intersect in a single point called the incenter. Thispoint is the center of a circle that just touches the three sides of the triangle. Except for thethree points where the circle touches the sides, the circle is inside the triangle. The circle issaid to be inscribed in the triangle.
1. With a compass and a straightedge, construct the inscribed circle for "PQR by following the steps below.Step 1 Construct the bisectors of ! P and ! Q. Label the point
where the bisectors meet A.Step 2 Construct a perpendicular segment from A to R!Q!. Use
the letter B to label the point where the perpendicularsegment intersects R!Q!.
Step 3 Use a compass to draw the circle with center at A andradius A!B!.
Construct the inscribed circle in each triangle.
2. 3.
The three perpendicular bisectors of the sides of a triangle also meet in a single point. Thispoint is the center of the circumscribed circle, which passes through each vertex of thetriangle. Except for the three points where the circle touches the triangle, the circle isoutside the triangle.
4. Follow the steps below to construct the circumscribed circle for "FGH.Step 1 Construct the perpendicular bisectors of F!G! and F!H!.
Use the letter A to label the point where theperpendicular bisectors meet.
Step 2 Draw the circle that has center A and radius A!F!.
Construct the circumscribed circle for each triangle.
5. 6.
F H
G
P
QR
A
B
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
5-15-1
Study Guide and InterventionInequalities and Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
5-25-2
© Glencoe/McGraw-Hill 251 Glencoe Geometry
Less
on
5-2
Angle Inequalities Properties of inequalities, including the Transitive, Addition,Subtraction, Multiplication, and Division Properties of Inequality, can be used withmeasures of angles and segments. There is also a Comparison Property of Inequality.
For any real numbers a and b, either a % b, a ! b, or a & b.
The Exterior Angle Theorem can be used to prove this inequality involving an exterior angle.
If an angle is an exterior angle of aExterior Angle triangle, then its measure is greater than Inequality Theorem the measure of either of its corresponding
remote interior angles.
m!1 & m!A, m!1 & m!B
List all angles of "EFG whose measures are less than m!1.The measure of an exterior angle is greater than the measure of either remote interior angle. So m!3 % m!1 and m!4 % m!1.
List all angles that satisfy the stated condition.
1. all angles whose measures are less than m!1
2. all angles whose measures are greater than m!3
3. all angles whose measures are less than m!1
4. all angles whose measures are greater than m!1
5. all angles whose measures are less than m!7
6. all angles whose measures are greater than m!2
7. all angles whose measures are greater than m!5
8. all angles whose measures are less than m!4
9. all angles whose measures are less than m!1
10. all angles whose measures are greater than m!4
R O
Q
N
P3 456
Exercises 9–10
78
21
S
X T W V
3
4
5
67 2 1
U
Exercises 3–8
M J K
3
4 521
L
Exercises 1–2
H E F3
4
21
G
A C D1
B
ExampleExample
© Glencoe/McGraw-Hill 252 Glencoe Geometry
Angle-Side Relationships When the sides of triangles are not congruent, there is a relationship between the sides and angles of the triangles.
• If one side of a triangle is longer than another side, then the angle opposite the longer side has a greater measure than the If AC & AB, then m!B & m!C.angle opposite the shorter side. If m!A & m!C, then BC & AB.
• If one angle of a triangle has a greater measure than another angle, then the side opposite the greater angle is longer than the side opposite the lesser angle.
B C
A
Study Guide and Intervention (continued)
Inequalities and Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
5-25-2
List the angles in orderfrom least to greatest measure.
!T, !R, !S
R T9 cm
6 cm 7 cm
S
List the sides in orderfrom shortest to longest.
C!B!, A!B!, A!C!A B
C
20$
35$
125$
Example 1Example 1 Example 2Example 2
ExercisesExercises
List the angles or sides in order from least to greatest measure.
1. 2. 3.
Determine the relationship between the measures of the given angles.
4. !R, !RUS
5. !T, !UST
6. !UVS, !R
Determine the relationship between the lengths of the given sides.
7. A!C!, B!C!
8. B!C!, D!B!
9. A!C!, D!B!
A B
C
D30$
30$30$
90$
R V S
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24 24
22
21.635
A C
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4.0R T
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40$T S
R48 cm
23.7 cm
35 cm
Skills PracticeInequalities and Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
5-25-2
© Glencoe/McGraw-Hill 253 Glencoe Geometry
Less
on
5-2
Determine which angle has the greatest measure.
1. !1, !3, !4 2. !4, !5, !7
3. !2, !3, !6 4. !5, !6, !8
Use the Exterior Angle Inequality Theorem to list all angles that satisfy the stated condition.
5. all angles whose measures are less than m!1
6. all angles whose measures are less than m!9
7. all angles whose measures are greater than m!5
8. all angles whose measures are greater than m!8
Determine the relationship between the measures of the given angles.
9. m!ABD, m!BAD 10. m!ADB, m!BAD
11. m!BCD, m!CDB 12. m!CBD, m!CDB
Determine the relationship between the lengths of the given sides.
13. L!M!, L!P! 14. M!P!, M!N!
15. M!N!, N!P! 16. M!P!, L!P!
83$ 57$79$
44$59$
38$LN
P
M
2334
4139
35A
B C
D
1
2 4
6
7
8 93 5
1 2 4 6 7 8
35
© Glencoe/McGraw-Hill 254 Glencoe Geometry
Determine which angle has the greatest measure.
1. !1, !3, !4 2. !4, !8, !9
3. !2, !3, !7 4. !7, !8, !10
Use the Exterior Angle Inequality Theorem to list all angles that satisfy the stated condition.
5. all angles whose measures are less than m!1
6. all angles whose measures are less than m!3
7. all angles whose measures are greater than m!7
8. all angles whose measures are greater than m!2
Determine the relationship between the measures of the given angles.
9. m!QRW, m!RWQ 10. m!RTW, m!TWR
11. m!RST, m!TRS 12. m!WQR, m!QRW
Determine the relationship between the lengths of the given sides.
13. D!H!, G!H! 14. D!E!, D!G!
15. E!G!, F!G! 16. D!E!, E!G!
17. SPORTS The figure shows the position of three trees on one part of a Frisbee™ course. At which tree position is the angle between the trees the greatest?
53 ft
40 ft
3
2
1
37.5 ft
120$32$
48$ 113$
17$H
D E F
G
3447
45
44
22
14
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R
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12
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12
4 678 9
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3
5
Practice Inequalities and Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
5-25-2
Reading to Learn MathematicsInequalities and Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
5-25-2
© Glencoe/McGraw-Hill 255 Glencoe Geometry
Less
on
5-2
Pre-Activity How can you tell which corner is bigger?
Read the introduction to Lesson 5-2 at the top of page 247 in your textbook.
• Which side of the patio is opposite the largest corner?
• Which side of the patio is opposite the smallest corner?
Reading the Lesson1. Name the property of inequality that is illustrated by each of the following.
a. If x & 8 and 8 & y, then x & y.b. If x % y, then x # 7.5 % y # 7.5.c. If x & y, then #3x % #3y.d. If x is any real number, x & 0, x ! 0, or x % 0.
2. Use the definition of inequality to write an equation that shows that each inequality is true.a. 20 & 12 b. 101 & 99c. 8 & #2 d. 7 & #7e. #11 & #12 f. #30 & #45
3. In the figure, m!IJK ! 45 and m!H & m!I.a. Arrange the following angles in order from largest to
smallest: !I, !IJK, !H, !IJHb. Arrange the sides of "HIJ in order from shortest to longest.
c. Is "HIJ an acute, right, or obtuse triangle? Explain your reasoning.
d. Is "HIJ scalene, isosceles, or equilateral? Explain your reasoning.
Helping You Remember4. A good way to remember a new geometric theorem is to relate it to a theorem you
learned earlier. Explain how the Exterior Angle Inequality Theorem is related to theExterior Angle Theorem, and why the Exterior Angle Inequality Theorem must be true ifthe Exterior Angle Theorem is true.
KJH
I
© Glencoe/McGraw-Hill 256 Glencoe Geometry
Construction ProblemThe diagram below shows segment AB adjacent to a closed region. Theproblem requires that you construct another segment XY to the right of theclosed region such that points A, B, X, and Y are collinear. You are not allowedto touch or cross the closed region with your compass or straightedge.
Follow these instructions to construct a segment XY so that it iscollinear with segment AB.
1. Construct the perpendicular bisector of A!B!. Label the midpoint as point C,and the line as m.
2. Mark two points P and Q on line m that lie well above the closed region.Construct the perpendicular bisector n of P!Q!. Label the intersection oflines m and n as point D.
3. Mark points R and S on line n that lie well to the right of the closedregion. Construct the perpendicular bisector k of R!S!. Label theintersection of lines n and k as point E.
4. Mark point X on line k so that X is below line n and so that E!X! iscongruent to D!C!.
5. Mark points T and V on line k and on opposite sides of X, so that X!T! andX!V! are congruent. Construct the perpendicular bisector ! of T!V!. Call thepoint where the line ! hits the boundary of the closed region point Y. X!Y!corresponds to the new road.
Q
P
m
k
nD
R E
TX
V
BAC
S
ExistingRoad
Closed Region(Lake)
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
5-25-2
Study Guide and InterventionIndirect Proof
NAME ______________________________________________ DATE ____________ PERIOD _____
5-35-3
© Glencoe/McGraw-Hill 257 Glencoe Geometry
Less
on
5-3
Indirect Proof with Algebra One way to prove that a statement is true is to assumethat its conclusion is false and then show that this assumption leads to a contradiction ofthe hypothesis, a definition, postulate, theorem, or other statement that is accepted as true.That contradiction means that the conclusion cannot be false, so the conclusion must betrue. This is known as indirect proof.
Steps for Writing an Indirect Proof
1. Assume that the conclusion is false.2. Show that this assumption leads to a contradiction.3. Point out that the assumption must be false, and therefore, the conclusion must be true.
Given: 3x ! 5 % 8Prove: x % 1
Step 1 Assume that x is not greater than 1. That is, x ! 1 or x % 1.Step 2 Make a table for several possibilities for x ! 1 or x % 1. The
contradiction is that when x ! 1 or x % 1, then 3x " 5 is notgreater than 8.
Step 3 This contradicts the given information that 3x " 5 & 8. Theassumption that x is not greater than 1 must be false, which means that the statement “x & 1” must be true.
Write the assumption you would make to start an indirect proof of each statement.
1. If 2x & 14, then x & 7.
2. For all real numbers, if a " b & c, then a & c # b.
Complete the proof.Given: n is an integer and n2 is even.Prove: n is even.
3. Assume that
4. Then n can be expressed as 2a " 1 by
5. n2 ! Substitution
6. ! Multiply.
7. ! Simplify.
8. ! 2(2a2 " 2a) " 1
9. 2(2a2 " 2a)" 1 is an odd number. This contradicts the given that n2 is even,
so the assumption must be
10. Therefore,
x 3x " 5
1 8
0 5
#1 2
#2 #1
#3 #4
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 258 Glencoe Geometry
Indirect Proof with Geometry To write an indirect proof in geometry, you assumethat the conclusion is false. Then you show that the assumption leads to a contradiction.The contradiction shows that the conclusion cannot be false, so it must be true.
Given: m!C # 100Prove: !A is not a right angle.
Step 1 Assume that !A is a right angle.
Step 2 Show that this leads to a contradiction. If !A is a right angle,then m!A ! 90 and m!C " m!A ! 100 " 90 ! 190. Thus the sum of the measures of the angles of "ABC is greater than 180.
Step 3 The conclusion that the sum of the measures of the angles of "ABC is greater than 180 is a contradiction of a known property.The assumption that !A is a right angle must be false, which means that the statement “!A is not a right angle” must be true.
Write the assumption you would make to start an indirect proof of eachstatement.
1. If m!A ! 90, then m!B ! 45.
2. If A!V! is not congruent to V!E!, then "AVE is not isosceles.
Complete the proof.
Given: !1 " !2 and D!G! is not congruent to F!G!.Prove: D!E! is not congruent to F!E!.
3. Assume that Assume the conclusion is false.
4. E!G! " E!G!
5. "EDG " "EFG
6.
7. This contradicts the given information, so the assumption must
be
8. Therefore,
12
D G
FE
A B
C
Study Guide and Intervention (continued)
Indirect Proof
NAME ______________________________________________ DATE ____________ PERIOD _____
5-35-3
ExercisesExercises
ExampleExample
Skills PracticeIndirect Proof
NAME ______________________________________________ DATE ____________ PERIOD _____
5-35-3
© Glencoe/McGraw-Hill 259 Glencoe Geometry
Less
on
5-3
Write the assumption you would make to start an indirect proof of each statement.
1. m!ABC % m!CBA
2. "DEF " "RST
3. Line a is perpendicular to line b.
4. !5 is supplementary to !6.
PROOF Write an indirect proof.
5. Given: x2 " 8 ' 12Prove: x ' 2
6. Given: !D # !F.Prove: DE ( EF
D F
E
© Glencoe/McGraw-Hill 260 Glencoe Geometry
Write the assumption you would make to start an indirect proof of each statement.
1. B!D! bisects !ABC.
2. RT ! TS
PROOF Write an indirect proof.
3. Given: #4x " 2 % #10Prove: x & 3
4. Given: m!2 " m!3 ( 180Prove: a ⁄|| b
5. PHYSICS Sound travels through air at about 344 meters per second when thetemperature is 20°C. If Enrique lives 2 kilometers from the fire station and it takes 5 seconds for the sound of the fire station siren to reach him, how can you proveindirectly that it is not 20°C when Enrique hears the siren?
12
3
a
b
Practice Indirect Proof
NAME ______________________________________________ DATE ____________ PERIOD _____
5-35-3
Reading to Learn MathematicsIndirect Proof
NAME ______________________________________________ DATE ____________ PERIOD _____
5-35-3
© Glencoe/McGraw-Hill 261 Glencoe Geometry
Less
on
5-3
Pre-Activity How is indirect proof used in literature?
Read the introduction to Lesson 5-3 at the top of page 255 in your textbook.
How could the author of a murder mystery use indirect reasoning to showthat a particular suspect is not guilty?
Reading the Lesson1. Supply the missing words to complete the list of steps involved in writing an indirect proof.
Step 1 Assume that the conclusion is .
Step 2 Show that this assumption leads to a of the
or some other fact, such as a definition, postulate,
, or corollary.
Step 3 Point out that the assumption must be and, therefore, the
conclusion must be .
2. State the assumption that you would make to start an indirect proof of each statement.
a. If #6x & 30, then x % #5.
b. If n is a multiple of 6, then n is a multiple of 3.
c. If a and b are both odd, then ab is odd.
d. If a is positive and b is negative, then ab is negative.
e. If F is between E and D, then EF " FD ! ED.
f. In a plane, if two lines are perpendicular to the same line, then they are parallel.
g. Refer to the figure. h. Refer to the figure.
If AB ! AC, then m!B ! m!C. In "PQR, PR " QR & QP.
Helping You Remember3. A good way to remember a new concept in mathematics is to relate it to something you have
already learned. How is the process of indirect proof related to the relationship between aconditional statement and its contrapositive?
P
RQ
A C
B
© Glencoe/McGraw-Hill 262 Glencoe Geometry
More CounterexamplesSome statements in mathematics can be proven false by counterexamples.Consider the following statement.
For any numbers a and b, a # b ! b # a.
You can prove that this statement is false in general if you can find oneexample for which the statement is false.
Let a ! 7 and b ! 3. Substitute these values in the equation above.
7 # 3 # 3 # 74 ( #4
In general, for any numbers a and b, the statement a # b ! b # a is false.You can make the equivalent verbal statement: subtraction is not acommutative operation.
In each of the following exercises a, b, and c are any numbers. Prove that the statement is false by counterexample.
1. a # (b # c) # (a # b) # c 2. a ) (b ) c) # (a ) b) ) c
3. a ) b # b ) a 4. a ) (b " c) # (a ) b) " (a ) c)
5. a " (bc) # (a " b)(a " c) 6. a2 " a2 # a4
7. Write the verbal equivalents for Exercises 1, 2, and 3.
8. For the Distributive Property a(b " c) ! ab " ac it is said that multiplicationdistributes over addition. Exercises 4 and 5 prove that some operations do notdistribute. Write a statement for each exercise that indicates this.
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
5-35-3
Study Guide and InterventionThe Triangle Inequality
NAME ______________________________________________ DATE ____________ PERIOD _____
5-45-4
© Glencoe/McGraw-Hill 263 Glencoe Geometry
Less
on
5-4
The Triangle Inequality If you take three straws of lengths 8 inches, 5 inches, and 1 inch and try to make a triangle with them, you will find that it is not possible. Thisillustrates the Triangle Inequality Theorem.
Triangle Inequality The sum of the lengths of any two sides of aTheorem triangle is greater than the length of the third side.
The measures of two sides of a triangle are 5 and 8. Find a rangefor the length of the third side.By the Triangle Inequality, all three of the following inequalities must be true.
5 " x & 8 8 " x & 5 5 " 8 & xx & 3 x & #3 13 & x
Therefore x must be between 3 and 13.
Determine whether the given measures can be the lengths of the sides of atriangle. Write yes or no.
1. 3, 4, 6 2. 6, 9, 15
3. 8, 8, 8 4. 2, 4, 5
5. 4, 8, 16 6. 1.5, 2.5, 3
Find the range for the measure of the third side given the measures of two sides.
7. 1 and 6 8. 12 and 18
9. 1.5 and 5.5 10. 82 and 8
11. Suppose you have three different positive numbers arranged in order from least togreatest. What single comparison will let you see if the numbers can be the lengths ofthe sides of a triangle?
BC
A
a
cb
ExercisesExercises
ExampleExample
© Glencoe/McGraw-Hill 264 Glencoe Geometry
Distance Between a Point and a Line
Study Guide and Intervention (continued)
The Triangle Inequality
NAME ______________________________________________ DATE ____________ PERIOD _____
5-45-4
The perpendicular segment from a point toa line is the shortest segment from thepoint to the line.
P!C! is the shortest segment from P to AB#!".
The perpendicular segment from a point toa plane is the shortest segment from thepoint to the plane.
Q!T! is the shortest segment from Q to plane N .
Q
TN
B
P
CA
Given: Point P is equidistant from the sides of an angle.
Prove: B!A! " C!A!Proof:1. Draw B!P! and C!P! ⊥ to 1. Dist. is measured
the sides of !RAS. along a ⊥.2. !PBA and !PCA are right angles. 2. Def. of ⊥ lines3. "ABP and "ACP are right triangles. 3. Def. of rt. "
4. !PBA " !PCA 4. Rt. angles are ".5. P is equidistant from the sides of !RAS. 5. Given6. B!P! " C!P! 6. Def. of equidistant7. A!P! " A!P! 7. Reflexive Property8. "ABP " "ACP 8. HL9. B!A! " C!A! 9. CPCTC
Complete the proof.Given: "ABC " "RST; !D " !UProve: A!D! " R!U!Proof:
1. "ABC " "RST; !D " !U 1.
2. A!C! " R!T! 2.
3. !ACB " !RTS 3.
4. !ACB and !ACD are a linear pair; 4. Def. of !RTS and !RTU are a linear pair.
5. !ACB and !ACD are supplementary; 5.!RTS and !RTU are supplementary.
6. 6. Angles suppl. to " angles are ".
7. "ADC " "RUT 7.
8. 8. CPCTC
A
D C B
R
U T S
AS C
PB
R
ExampleExample
ExercisesExercises
Skills PracticeThe Triangle Inequality
NAME ______________________________________________ DATE ____________ PERIOD _____
5-45-4
© Glencoe/McGraw-Hill 265 Glencoe Geometry
Less
on
5-4
Determine whether the given measures can be the lengths of the sides of atriangle. Write yes or no.
1. 2, 3, 4 2. 5, 7, 9
3. 4, 8, 11 4. 13, 13, 26
5. 9, 10, 20 6. 15, 17, 19
7. 14, 17, 31 8. 6, 7, 12
Find the range for the measure of the third side of a triangle given the measuresof two sides.
9. 5 and 9 10. 7 and 14
11. 8 and 13 12. 10 and 12
13. 12 and 15 14. 15 and 27
15. 17 and 28 16. 18 and 22
ALGEBRA Determine whether the given coordinates are the vertices of a triangle.Explain.
17. A(3, 5), B(4, 7), C(7, 6) 18. S(6, 5), T(8, 3), U(12, #1)
19. H(#8, 4), I(#4, 2), J(4, #2) 20. D(1, #5), E(#3, 0), F(#1, 0)
© Glencoe/McGraw-Hill 266 Glencoe Geometry
Determine whether the given measures can be the lengths of the sides of atriangle. Write yes or no.
1. 9, 12, 18 2. 8, 9, 17
3. 14, 14, 19 4. 23, 26, 50
5. 32, 41, 63 6. 2.7, 3.1, 4.3
7. 0.7, 1.4, 2.1 8. 12.3, 13.9, 25.2
Find the range for the measure of the third side of a triangle given the measuresof two sides.
9. 6 and 19 10. 7 and 29
11. 13 and 27 12. 18 and 23
13. 25 and 38 14. 31 and 39
15. 42 and 6 16. 54 and 7
ALGEBRA Determine whether the given coordinates are the vertices of a triangle.Explain.
17. R(1, 3), S(4, 0), T(10, #6) 18. W(2, 6), X(1, 6), Y(4, 2)
19. P(#3, 2), L(1, 1), M(9, #1) 20. B(1, 1), C(6, 5), D(4, #1)
21. GARDENING Ha Poong has 4 lengths of wood from which he plans to make a border for atriangular-shaped herb garden. The lengths of the wood borders are 8 inches, 10 inches,12 inches, and 18 inches. How many different triangular borders can Ha Poong make?
Practice The Triangle Inequality
NAME ______________________________________________ DATE ____________ PERIOD _____
5-45-4
Reading to Learn MathematicsThe Triangle Inequality
NAME ______________________________________________ DATE ____________ PERIOD _____
5-45-4
© Glencoe/McGraw-Hill 267 Glencoe Geometry
Less
on
5-4
Pre-Activity How can you use the Triangle Inequality Theorem when traveling?
Read the introduction to Lesson 5-4 at the top of page 261 in your textbook.
In addition to the greater distance involved in flying from Chicago toColumbus through Indianapolis rather than flying nonstop, what are twoother reasons that it would take longer to get to Columbus if you take twoflights rather than one?
Reading the Lesson
1. Refer to the figure.
Which statements are true?A. DE & EF " FD B. DE ! EF " FDC. EG ! EF " FG D. ED " DG & EGE. The shortest distance from D to EG#!" is DF.F. The shortest distance from D to EG#!" is DG.
2. Complete each sentence about "XYZ.
a. If XY ! 8 and YZ ! 11, then the range of values for XZ is % XZ % .
b. If XY ! 13 and XZ ! 25, then YZ must be between and .
c. If "XYZ is isosceles with !Z as the vertex angle, and XZ ! 8.5, then the range of
values for XY is % XY % .
d. If XZ ! a and YZ ! b, with b % a, then the range for XY is % XY % .
Helping You Remember
3. A good way to remember a new theorem is to state it informally in different words. Howcould you restate the Triangle Inequality Theorem?
ZX
Y
G
D
EF
© Glencoe/McGraw-Hill 268 Glencoe Geometry
Constructing TrianglesThe measurements of the sides of a triangle are given. If a triangle having sideswith these measurements is not possible, then write impossible. If a triangle ispossible, draw it and measure each angle with a protractor.
1. AR ! 5 cm m!A ! 2. PI ! 8 cm m!P !
RT ! 3 cm m!R ! IN ! 3 cm m!I !
AT ! 6 cm m!T ! PN ! 2 cm m!N !
3. ON ! 10 cm m!O ! 4. TW ! 6 cm m!T !
NE ! 5.3 cm m!N ! WO ! 7 cm m!W!
GE ! 4.6 cm m!E ! TO ! 2 cm m!O !
5. BA ! 3.l cm m!B ! 6. AR ! 4 cm m!A !
AT ! 8 cm m!A ! RM ! 5 cm m!R !
BT ! 5 cm m!T ! AM ! 3 cm m!M !
M
RAT
BA
W
T
O
A R
T
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
5-45-4
Study Guide and InterventionInequalities Involving Two Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
5-55-5
© Glencoe/McGraw-Hill 269 Glencoe Geometry
Less
on
5-5
SAS Inequality The following theorem involves the relationship between the sides oftwo triangles and an angle in each triangle.
If two sides of a triangle are congruent to two sides of another triangle and the included angle in one triangle has a
SAS Inequality/Hinge Theorem greater measure than the included angle in the other, then the third side of the If R!S! " A!B!, S!T! " B!C!, andfirst triangle is longer than the third side m!S & m!B, then RT & AC.of the second triangle.
Write an inequality relating the lengths of C!D! and A!D!.Two sides of "BCD are congruent to two sides of "BAD and m!CBD & m!ABD. By the SAS Inequality/Hinge Theorem,CD & AD.
Write an inequality relating the given pair of segment measures.
1. 2.
MR, RP AD, CD
3. 4.
EG, HK MR, PR
Write an inequality to describe the possible values of x.
5. 6.
62$65$
2.7 cm1.8 cm
1.8 cm (3x " 2.1) cm
115$120$ 24 cm
24 cm40 cm
(4x " 10) cm
M R
N P
48$46$
20 25
20
E G
H K
J
F60$
62$
10
10
42
42
C
A
DB
22$
38$
N
R
P
M
21$
19$
B D
A
28$22$
C
S T80$
R
B C60$
A
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 270 Glencoe Geometry
SSS Inequality The converse of the Hinge Theorem is also useful when two triangleshave two pairs of congruent sides.
If two sides of a triangle are congruent to two sidesof another triangle and the third side in one triangle
SSS Inequalityis longer than the third side in the other, then the angle between the pair of congruent sides in the first triangle is greater than the corresponding angle in the second triangle. If NM ! SR, MP ! RT, and NP & ST, then
m!M & m!R.
Write an inequality relating the measures of !ABD and !CBD.Two sides of "ABD are congruent to two sides of "CBD, and AD & CD.By the SSS Inequality, m!ABD & m!CBD.
Write an inequality relating the given pair of angle measures.
1. 2.
m!MPR, m!NPR m!ABD, m!CBD
3. 4.
m!C, m!Z m!XYW, m!WYZ
Write an inequality to describe the possible values of x.
5. 6.
33$
60 cm
60 cm
36 cm
30 cm(3x " 3)$
(1–2x " 6)$
52$30
30
28
12
42
28
ZW
XY
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A X
B30
5048 24
24Z Y
11 16
2626
B
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13
10
M
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NP
13
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3838
2323 3336
TR
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Study Guide and Intervention (continued)
Inequalities Involving Two Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
5-55-5
ExampleExample
ExercisesExercises
Skills PracticeInequalities Involving Two Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
5-55-5
© Glencoe/McGraw-Hill 271 Glencoe Geometry
Less
on
5-5
Write an inequality relating the given pair of angles or segment measures.
1. m!BXA, m!DXA
2. BC, DC
Write an inequality relating the given pair of angles or segment measures.
3. m!STR, m!TRU 4. PQ, RQ
5. In the figure, B!A!, B!D!, B!C!, and B!E! are congruent and AC % DE.How does m!1 compare with m!3? Explain your thinking.
6. Write a two-column proof.Given: B!A! " D!A!
BC & DCProve: m!1 & m!2
12
B
A
D
C
12
3
B
AD C
E
95$7 7
85$P RS
Q31
30
22 22
R S
U T
6
98
3
3
B
A C
D
X
© Glencoe/McGraw-Hill 272 Glencoe Geometry
Write an inequality relating the given pair of angles or segment measures.
1. AB, BK 2. ST, SR
3. m!CDF, m!EDF 4. m!R, m!T
5. Write a two-column proof.Given: G is the midpoint of D!F!.
m!1 & m!2Prove: ED & EF
6. TOOLS Rebecca used a spring clamp to hold together a chair leg she repaired with wood glue. When she opened the clamp,she noticed that the angle between the handles of the clampdecreased as the distance between the handles of the clampdecreased. At the same time, the distance between the gripping ends of the clamp increased. When she released the handles, the distance between the gripping end of the clamp decreased and the distance between the handles increased.Is the clamp an example of the SAS or SSS Inequality?
1 2D F
E
G
20 21
R TS
J K
14 14
14
13
12C F
ED
(x ! 3)$(x " 3)$
10 10
R TS
Q
40$
30$
60$A KM
B
Practice Inequalities Involving Two Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
5-55-5
Reading to Learn MathematicsInequalities Involving Two Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
5-55-5
© Glencoe/McGraw-Hill 273 Glencoe Geometry
Less
on
5-5
Pre-Activity How does a backhoe work?
Read the introduction to Lesson 5-5 at the top of page 267 in your textbook.
What is the main kind of task that backhoes are used to perform?
Reading the Lesson1. Refer to the figure. Write a conclusion that you can draw from the given information.
Then name the theorem that justifies your conclusion.
a. L!M! " O!P!, M!N! " P!Q!, and LN & OQ
b. L!M! " O!P!, M!N! " P!Q!, and m!P % m!M
c. LM ! 8, LN ! 15, OP ! 8, OQ ! 15, m!L ! 22, and m!O ! 21
2. In the figure, "EFG is isosceles with base F!G! and F is the midpoint of D!G!. Determine whether each of the following is a valid conclusion that you can draw based on the given information. (Write valid or invalid.) If the conclusion is valid,identify the definition, property, postulate, or theorem that supports it.
a. !3 " !4
b. DF ! GF
c. "DEF is isosceles.
d. m!3 & m!1
e. m!2 & m!4
f. m!2 & m!3
g. DE & EG
h. DE & FG
Helping You Remember3. A good way to remember something is to think of it in concrete terms. How can you
illustrate the Hinge Theorem with everyday objects?
F GD
E
1 2 3 4
N Q PM
L O
© Glencoe/McGraw-Hill 274 Glencoe Geometry
Drawing a DiagramIt is useful and often necessary to draw a diagram of the situationbeing described in a problem. The visualization of the problem ishelpful in the process of problem solving.
The roads connecting the towns of Kings,Chana, and Holcomb form a triangle. Davis Junction islocated in the interior of this triangle. The distances fromDavis Junction to Kings, Chana, and Holcomb are 3 km,4 km, and 5 km, respectively. Jane begins at Holcomb anddrives directly to Chana, then to Kings, and then back toHolcomb. At the end of her trip, she figures she has traveled25 km altogether. Has she figured the distance correctly?
To solve this problem, a diagram can be drawn. Based on this diagram and the Triangle Inequality Theorem, the distance from Holcomb to Chana is less than 9 km. Similarly,the distance from Chana to Kings is less than 7 km, and thedistance from Kings to Holcomb is less than 8 km.
Therefore, Jane must have traveled less than (9 " 7 " 8) km or 24 km versus her calculated distance of 25 km.
Explain why each of the following statements is true.Draw and label a diagram to be used in the explanation.
1. If an altitude is drawn to one side of a triangle, then thelength of the altitude is less than one-half the sum of thelengths of the other two sides.
2. If point Q is in the interior of *ABC and on the angle bisectorof !B, then Q is equidistant from A!B! and C!B!. (Hint: Draw Q!D!and Q!E! such that Q!D! $ A!B! and Q!E! $ C!B!.)
C E B
A
Q
D
A D C
B
Kings
DavisJunction
Chana Holcomb
3 km
5 km4 km
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
5-55-5
ExampleExample
© Glencoe/McGraw-Hill A2 Glencoe Geometry
Stu
dy
Gu
ide
and I
nte
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tion
Bis
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____
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5-1
5-1
©G
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5G
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Lesson 5-1
Perp
end
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Bis
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An
gle
Bis
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per
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dic
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asi
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sth
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s m
idpo
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ther
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line
is a
n an
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bise
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,whi
chdi
vide
s an
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to t
wo
cong
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gles
.
Tw
o pr
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of
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are:
(1) a
poi
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or o
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if it
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and
(2) t
he t
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per
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of t
he s
ides
of a
tri
angl
e m
eet
at a
poi
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eci
rcu
mce
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the
tri
angl
e,th
at is
equ
idis
tant
fro
m t
he t
hree
ver
tice
s of
the
tri
angl
e.
Tw
o pr
oper
ties
of
angl
e bi
sect
ors
are:
(1) a
poi
nt is
on
the
angl
e bi
sect
or o
f an
angl
e if
and
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y if
it is
equ
idis
tant
from
the
sid
esof
the
ang
le,a
nd(2
) the
thr
ee a
ngle
bis
ecto
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f a t
rian
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a p
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led
the
ince
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the
tria
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,tha
t is
equ
idis
tant
fro
m t
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hree
sid
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f th
e tr
iang
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BD
! !"
is t
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rbi
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C!.F
ind
x.
BD
!!"
is t
he p
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ular
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A!C!
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3x"
8 !
5x#
614
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7 !
x
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8
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MR
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is t
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if m
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and
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1 !
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2.5x
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24 !
3x8
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3.
DE
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Fis
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DF
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.bi
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or o
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.x
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;y#
2x
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4.Fo
r w
hat
kind
s of
tri
angl
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can
the
per
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sid
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so b
e an
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f th
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gle
oppo
site
the
sid
e?is
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les
tria
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,equ
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ral t
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gle
5.Fo
r w
hat
kind
of
tria
ngle
do
the
perp
endi
cula
r bi
sect
ors
inte
rsec
t in
a p
oint
out
side
the
tria
ngle
?ob
tuse
tri
angl
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FE
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Med
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line
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the
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the
mid
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the
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ters
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Theo
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Poi
nts
R,S
,an
d T
are
the
mid
poi
nts
of
A !B!
,B!C!
and
A!C!
,res
pec
tive
ly.F
ind
x,y
,an
d z
.
CU
!$2 3$ C
RB
U!
$2 3$ BT
AU
!$2 3$ A
S
6x!
$2 3$ (6x
"15
)24
!$2 3$ (
24 "
3y#
3)6z
"4
!$2 3$ (
6z"
4 "
11)
9x!
6x"
1536
!24
"3y
#3
$3 2$ (6z
"4)
!6z
"4
"11
3x!
1536
!21
"3y
9z"
6!
6z"
15x
!5
15!
3y3z
!9
5!
yz
!3
Fin
d t
he
valu
e of
eac
h v
aria
ble.
1.x
#4
2.x
#6;
y#
5
B!D!
is a
med
ian.
AB
!C
B;D
,E,a
nd F
are
mid
poin
ts.
3.x
#3;
y#
54.
x#
12;y
#5;
z#
2
EH
!F
H!
HG
5.x
#2;
y#
2;z
#2
6.x
#6;
y#
5;z
#8
Vis
the
cen
troi
d of
"R
ST
;D
is t
he c
entr
oid
of "
AB
C.
TP
!18
;MS
!15
;RN
!24
7.Fo
r w
hat
kind
of
tria
ngle
are
the
med
ians
and
ang
le b
isec
tors
the
sam
e se
gmen
ts?
equi
late
ral t
rian
gle
8.Fo
r w
hat
kind
of
tria
ngle
is t
he c
entr
oid
outs
ide
the
tria
ngle
?no
t po
ssib
le
P
M
V
T
N
RS
y x
z
G
FE
B
AC
24
329z
! 6
6z6x
8y
MJ
PN
O
LK3y
! 5
2 x6z12
24 10H
GF
E
7x !
4
9x "
2 5y
DB
EF
AC
9x !
6
10x
3y15D
BA
C
6x !
3
7x "
1
AC
T
SR
U
B 3y "
3
6x
1524
11
6z !
4
AC
F
ED
L
Bce
ntro
id
Stu
dy
Gu
ide
and I
nte
rven
tion
(con
tinu
ed)
Bis
ecto
rs,M
edia
ns,a
nd A
ltitu
des
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
5-1
5-1
Exam
ple
Exam
ple
Exercis
esExercis
es
Answers (Lesson 5-1)
© Glencoe/McGraw-Hill A3 Glencoe Geometry
An
swer
s
Skil
ls P
ract
ice
Bis
ecto
rs,M
edia
ns,a
nd A
ltitu
des
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
5-1
5-1
©G
lenc
oe/M
cGra
w-H
ill24
7G
lenc
oe G
eom
etry
Lesson 5-1
ALG
EBR
AF
or E
xerc
ises
1–4
,use
th
e gi
ven
in
form
atio
n t
o fi
nd
eac
h v
alu
e.
1.F
ind
xif
E!G!
is a
med
ian
of "
DE
F.
2.F
ind
xan
d R
Tif
S!U!
is a
med
ian
of "
RS
T.
x#
9x
#18
;RT
#12
0
3.F
ind
xan
d E
Fif
B!D!
is a
n an
gle
bise
ctor
.4.
Fin
d x
and
IJif
H!K!
is a
n al
titu
de o
f "H
IJ.
x#
3.5;
EF
#13
x#
29;I
J#
57
ALG
EBR
AF
or E
xerc
ises
5–7
,use
th
e fo
llow
ing
info
rmat
ion
.In
"L
MN
,P,Q
,and
Rar
e th
e m
idpo
ints
of L !
M!,M!
N!,a
nd L!
N!,
resp
ecti
vely
.
5.F
ind
x.4
6.F
ind
y.0.
87.
Fin
d z.
0.7
ALG
EBR
AL
ines
a,b
,an
d c
are
per
pen
dic
ula
r bi
sect
ors
of "
PQ
Ran
d m
eet
at A
.
8.F
ind
x.1
9.F
ind
y.6
10.F
ind
z.2
CO
OR
DIN
ATE
GEO
MET
RYT
he
vert
ices
of
"H
IJar
e G
(1,0
),H
(6,0
),an
d I
(3,6
).F
ind
the
coor
din
ates
of
the
poi
nts
of
con
curr
ency
of
"H
IJ.
11.o
rtho
cent
er12
.cen
troi
d13
.cir
cum
cent
er
(3,1
)"&1 30 &
,2#
"&7 2& ,&5 2& #5y
" 6
8x !
16
7 z !
4
24
18
RQ
A
ab
c
P
y !
1
2z2.
8
23.
6
x
L
NQ
RB
P
M
( 3x
! 3
) $x !
8 x "
9
I
JH
K
AD4x
" 1
2x !
6B
G EF
C
RU 5x "
30
2x !
24
S
T
DG 3x !
1
5x "
17
E
F
©G
lenc
oe/M
cGra
w-H
ill24
8G
lenc
oe G
eom
etry
ALG
EBR
AIn
"A
BC
,B!F!
is t
he
angl
e bi
sect
or o
f !
AB
C,A!
E!,B!
F!,
and
C !D!
are
med
ian
s,an
d P
is t
he
cen
troi
d.
1.F
ind
xif
DP
!4x
#3
and
CP
!30
.4.
5
2.F
ind
yif
AP
!y
and
EP
!18
.36
3.F
ind
zif
FP
!5z
"10
and
BP
!42
.2.
2
4.If
m!
AB
C!
xan
d m
!B
AC
!m
!B
CA
!2x
#10
,is
B!F!
an a
ltit
ude?
Exp
lain
.Ye
s;si
nce
x#
40 a
nd B!
F!is
an
angl
e bi
sect
or,i
t fo
llow
s th
at m
!B
AF
#70
an
d m
!A
BF
#20
.So
m!
AFB
#90
,and
B!F!
⊥A!
C!.
ALG
EBR
AIn
"P
RS
,P!T!
is a
n a
ltit
ud
e an
d P!
X!is
a m
edia
n.
5.F
ind
RS
if R
X!
x"
7 an
d S
X!
3x#
11.
32
6.F
ind
RT
if R
T!
x#
6 an
d m
!P
TR
!8x
#6.
6
ALG
EBR
AIn
"D
EF
,G!I!
is a
per
pen
dic
ula
r bi
sect
or.
7.F
ind
xif
EH
!16
and
FH
!6x
#5.
3.5
8.F
ind
yif
EG
!3.
2y#
1 an
d F
G!
2y"
5.5
9.F
ind
zif
m!
EG
H!
12z.
7.5
CO
OR
DIN
ATE
GEO
MET
RYT
he
vert
ices
of
"S
TU
are
S(0
,1),
T(4
,7),
and
U(8
,"3)
.F
ind
th
e co
ord
inat
es o
f th
e p
oin
ts o
f co
ncu
rren
cy o
f "
ST
U.
10.o
rtho
cent
er11
.cen
troi
d12
.cir
cum
cent
er
"&5 4& ,&3 2& #
"4,&5 3& #
"&4 83 &,&
7 4& #or
(5.3
75,1
.75)
13.M
OB
ILES
Nab
uko
wan
ts t
o co
nstr
uct
a m
obile
out
of
flat
tri
angl
es s
o th
at t
he s
urfa
ces
of t
he t
rian
gles
han
g pa
ralle
l to
the
floo
r w
hen
the
mob
ile is
sus
pend
ed.H
ow c
anN
abuk
o be
cer
tain
tha
t sh
e ha
ngs
the
tria
ngle
s to
ach
ieve
thi
s ef
fect
?S
he n
eeds
to
hang
eac
h tr
iang
le f
rom
its
cent
er o
f gr
avity
or
cent
roid
,w
hich
is t
he p
oint
at
whi
ch t
he t
hree
med
ians
of
the
tria
ngle
inte
rsec
t.
DI
HF
G
E
SR
P TX
AC F
E DP
B
Pra
ctic
e (A
vera
ge)
Bis
ecto
rs,M
edia
ns,a
nd A
ltitu
des
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
5-1
5-1
Answers (Lesson 5-1)
© Glencoe/McGraw-Hill A4 Glencoe Geometry
Rea
din
g t
o L
earn
Math
emati
csB
isec
tors
,Med
ians
,and
Alti
tude
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
5-1
5-1
©G
lenc
oe/M
cGra
w-H
ill24
9G
lenc
oe G
eom
etry
Lesson 5-1
Pre-
Act
ivit
yH
ow c
an y
ou b
alan
ce a
pap
er t
rian
gle
on a
pen
cil
poi
nt?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 5-
1 at
the
top
of
page
238
in y
our
text
book
.
Dra
w a
ny t
rian
gle
and
conn
ect
each
ver
tex
to t
he m
idpo
int
of t
he o
ppos
ite
side
to
form
the
thr
ee m
edia
ns o
f th
e tr
iang
le.I
s th
e po
int
whe
re t
he t
hree
med
ians
inte
rsec
t th
e m
idpo
int
of e
ach
of t
he m
edia
ns?
Sam
ple
answ
er:
No;
the
inte
rsec
tion
poin
t ap
pear
s to
be
mor
e th
an h
alfw
ayfr
om e
ach
vert
ex t
o th
e m
idpo
int
of t
he o
ppos
ite s
ide.
Rea
din
g t
he
Less
on
1.U
nder
line
the
corr
ect
wor
d or
phr
ase
to c
ompl
ete
each
sen
tenc
e.
a.T
hree
or
mor
e lin
es t
hat
inte
rsec
t at
a c
omm
on p
oint
are
cal
led
(par
alle
l/per
pend
icul
ar/c
oncu
rren
t) li
nes.
b.A
ny p
oint
on
the
perp
endi
cula
r bi
sect
or o
f a
segm
ent
is
(par
alle
l to/
cong
ruen
t to
/equ
idis
tant
fro
m)
the
endp
oint
s of
the
seg
men
t.
c.A
(n)
(alt
itud
e/an
gle
bise
ctor
/med
ian/
perp
endi
cula
r bi
sect
or)
of a
tri
angl
e is
a
segm
ent
draw
n fr
om a
ver
tex
of t
he t
rian
gle
perp
endi
cula
r to
the
line
con
tain
ing
the
oppo
site
sid
e.
d.T
he p
oint
of c
oncu
rren
cy o
f the
thr
ee p
erpe
ndic
ular
bis
ecto
rs o
f a t
rian
gle
is c
alle
d th
e(o
rtho
cent
er/c
ircu
mce
nter
/cen
troi
d/in
cent
er).
e.A
ny p
oint
in t
he in
teri
or o
f an
ang
le t
hat
is e
quid
ista
nt f
rom
the
sid
es o
f th
at a
ngle
lie
s on
the
(m
edia
n/an
gle
bise
ctor
/alt
itud
e).
f.T
he p
oint
of
conc
urre
ncy
of t
he t
hree
ang
le b
isec
tors
of
a tr
iang
le is
cal
led
the
(ort
hoce
nter
/cir
cum
cent
er/c
entr
oid/
ince
nter
).
2.In
the
fig
ure,
Eis
the
mid
poin
t of
A !B!
,Fis
the
mid
poin
t of
B!C!
,an
d G
is t
he m
idpo
int
of A !
C!.
a.N
ame
the
alti
tude
s of
"A
BC
.A!
C!,B!
C!,C!
D!b.
Nam
e th
e m
edia
ns o
f "A
BC
.A!
F!,B!
G!,C!
E!c.
Nam
e th
e ce
ntro
id o
f "A
BC
.H
d.N
ame
the
orth
ocen
ter
of "
AB
C.
Ce.
If A
F!
12 a
nd C
E!
9,fi
nd A
Han
d H
E.
AH
#8,
HE
#3
Hel
pin
g Y
ou
Rem
emb
er
3.A
goo
d w
ay t
o re
mem
ber
som
ethi
ng is
to
expl
ain
it t
o so
meo
ne e
lse.
Supp
ose
that
acl
assm
ate
is h
avin
g tr
oubl
e re
mem
beri
ng w
heth
er t
he c
ente
r of
gra
vity
of
a tr
iang
le is
the
orth
ocen
ter,
the
cent
roid
,the
ince
nter
,or
the
circ
umce
nter
of
the
tria
ngle
.Sug
gest
aw
ay t
o re
mem
ber
whi
ch p
oint
it is
.S
ampl
e an
swer
:The
ter
ms
cent
roid
and
cent
er o
f gra
vity
mea
n th
e sa
me
thin
g an
d in
bot
h te
rms,
the
lett
ers
“cen
t”co
me
at t
he b
egin
ning
of
the
term
s.
AB
C
FG
ED
H
©G
lenc
oe/M
cGra
w-H
ill25
0G
lenc
oe G
eom
etry
Insc
ribe
d an
d C
ircu
msc
ribe
d C
ircl
esT
he t
hree
ang
le b
isec
tors
of
a tr
iang
le in
ters
ect
in a
sin
gle
poin
t ca
lled
the
ince
nte
r.T
his
poin
t is
the
cen
ter
of a
cir
cle
that
just
tou
ches
the
thr
ee s
ides
of
the
tria
ngle
.Exc
ept
for
the
thre
e po
ints
whe
re t
he c
ircl
e to
uche
s th
e si
des,
the
circ
le is
insi
de t
he t
rian
gle.
The
cir
cle
issa
id t
o be
insc
ribe
d in
the
tri
angl
e.
1.W
ith
a co
mpa
ss a
nd a
str
aigh
tedg
e,co
nstr
uct
the
insc
ribe
d ci
rcle
for
"P
QR
by f
ollo
win
g th
e st
eps
belo
w.
Ste
p 1
Con
stru
ct t
he b
isec
tors
of !
Pan
d !
Q.L
abel
the
poi
nt
whe
re t
he b
isec
tors
mee
t A
.S
tep
2C
onst
ruct
a p
erpe
ndic
ular
seg
men
t fr
om A
to R !
Q!.U
se
the
lett
er B
to la
bel t
he p
oint
whe
re t
he p
erpe
ndic
ular
segm
ent
inte
rsec
ts R !
Q!.
Ste
p 3
Use
a c
ompa
ss t
o dr
aw t
he c
ircl
e w
ith
cent
er a
t A
and
radi
us A !
B!.
Con
stru
ct t
he
insc
ribe
d c
ircl
e in
eac
h t
rian
gle.
2.3.
The
thr
ee p
erpe
ndic
ular
bis
ecto
rs o
f th
e si
des
of a
tri
angl
e al
so m
eet
in a
sin
gle
poin
t.T
his
poin
t is
the
cen
ter
of t
he c
ircu
msc
ribe
d ci
rcle
,whi
ch p
asse
s th
roug
h ea
ch v
erte
x of
the
tria
ngle
.Exc
ept
for
the
thre
e po
ints
whe
re t
he c
ircl
e to
uche
s th
e tr
iang
le,t
he c
ircl
e is
outs
ide
the
tria
ngle
.
4.Fo
llow
the
ste
ps b
elow
to
cons
truc
t th
e ci
rcum
scri
bed
circ
le
for
"F
GH
.S
tep
1C
onst
ruct
the
per
pend
icul
ar b
isec
tors
of F
!G!an
d F!H!
.U
se t
he le
tter
Ato
labe
l the
poi
nt w
here
the
perp
endi
cula
r bi
sect
ors
mee
t.S
tep
2D
raw
the
cir
cle
that
has
cen
ter
Aan
d ra
dius
A !F!.
Con
stru
ct t
he
circ
um
scri
bed
cir
cle
for
each
tri
angl
e.
5.6.
FH
G
AP
QR
A B
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
5-1
5-1
Answers (Lesson 5-1)
© Glencoe/McGraw-Hill A5 Glencoe Geometry
An
swer
s
Stu
dy
Gu
ide
and I
nte
rven
tion
Ineq
ualit
ies
and
Tria
ngle
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
5-2
5-2
©G
lenc
oe/M
cGra
w-H
ill25
1G
lenc
oe G
eom
etry
Lesson 5-2
An
gle
Ineq
ual
itie
sP
rope
rtie
s of
ineq
ualit
ies,
incl
udin
g th
e T
rans
itiv
e,A
ddit
ion,
Subt
ract
ion,
Mul
tipl
icat
ion,
and
Div
isio
n P
rope
rtie
s of
Ine
qual
ity,
can
be u
sed
wit
hm
easu
res
of a
ngle
s an
d se
gmen
ts.T
here
is a
lso
a C
ompa
riso
n P
rope
rty
of I
nequ
alit
y.
For
any
real
num
bers
aan
d b,
eith
er a
%b,
a!
b,or
a&
b.
The
Ext
erio
r A
ngle
The
orem
can
be
used
to
prov
e th
is in
equa
lity
invo
lvin
g an
ext
erio
r an
gle.
If an
ang
le is
an
exte
rior
angl
e of
aE
xter
ior
Ang
letr
iang
le, t
hen
its m
easu
re is
gre
ater
than
In
equa
lity
Theo
rem
the
mea
sure
of e
ither
of i
ts c
orre
spon
ding
re
mot
e in
terio
r an
gles
.
m!
1 &
m!
A, m
!1
&m
!B
Lis
t al
l an
gles
of
"E
FG
wh
ose
mea
sure
s ar
e le
ss t
han
m!
1.T
he m
easu
re o
f an
ext
erio
r an
gle
is g
reat
er t
han
the
mea
sure
of
eith
er r
emot
e in
teri
or a
ngle
.So
m!
3 %
m!
1 an
d m
!4
%m
!1.
Lis
t al
l an
gles
th
at s
atis
fy t
he
stat
ed c
ond
itio
n.
1.al
l ang
les
who
se m
easu
res
are
less
tha
n m
!1
!3,
!4
2.al
l ang
les
who
se m
easu
res
are
grea
ter
than
m!
3!
1,!
5
3.al
l ang
les
who
se m
easu
res
are
less
tha
n m
!1
!5,
!6
4.al
l ang
les
who
se m
easu
res
are
grea
ter
than
m!
1!
7
5.al
l ang
les
who
se m
easu
res
are
less
tha
n m
!7
!1,
!3,
!5,
!6,
!TU
V
6.al
l ang
les
who
se m
easu
res
are
grea
ter
than
m!
2!
4
7.al
l ang
les
who
se m
easu
res
are
grea
ter
than
m!
5!
1,!
7,!
TUV
8.al
l ang
les
who
se m
easu
res
are
less
tha
n m
!4
!2,
!3
9.al
l ang
les
who
se m
easu
res
are
less
tha
n m
!1
!4,
!5,
!7,
!N
PR
10.a
ll an
gles
who
se m
easu
res
are
grea
ter
than
m!
4!
1,!
8,!
OP
N,!
RO
Q
RO
QN
P3
456
Exer
cise
s 9–
10
78
21
S
XT
WV
3 4
5
67
21
U
Exer
cise
s 3–
8
MJ
K
3
45
21L Ex
erci
ses
1–2
HE
F3
4
21
G
AC
D1
B
Exam
ple
Exam
ple
©G
lenc
oe/M
cGra
w-H
ill25
2G
lenc
oe G
eom
etry
An
gle
-Sid
e R
elat
ion
ship
sW
hen
the
side
s of
tri
angl
es a
re
not
cong
ruen
t,th
ere
is a
rel
atio
nshi
p be
twee
n th
e si
des
and
angl
es o
f th
e tr
iang
les.
•If
one
sid
e of
a t
rian
gle
is lo
nger
tha
n an
othe
r si
de,t
hen
the
angl
e op
posi
te t
he lo
nger
sid
e ha
s a
grea
ter
mea
sure
tha
n th
e If
AC
&A
B, t
hen
m!
B&
m!
C.
angl
e op
posi
te t
he s
hort
er s
ide.
If m
!A
&m
!C
, the
n B
C&
AB
.
•If
one
ang
le o
f a
tria
ngle
has
a g
reat
er m
easu
re t
han
anot
her
angl
e,th
en t
he s
ide
oppo
site
the
gre
ater
ang
le is
long
er t
han
the
side
opp
osit
e th
e le
sser
ang
le.
BC
A
Stu
dy
Gu
ide
and I
nte
rven
tion
(con
tinu
ed)
Ineq
ualit
ies
and
Tria
ngle
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
5-2
5-2
Lis
t th
e an
gles
in
ord
erfr
om l
east
to
grea
test
mea
sure
.
!T
,!R
,!S
RT
9 cm
6 cm
7 cm
S
Lis
t th
e si
des
in
ord
erfr
om s
hor
test
to
lon
gest
.
C !B!
,A!B!
,A!C!
AB
C
20$
35$
125$
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exercis
esExercis
es
Lis
t th
e an
gles
or
sid
es i
n o
rder
fro
m l
east
to
grea
test
mea
sure
.
1.2.
3.
!T,
!R
,!S
R!S!,
S!T!,R!
T!!
C,!
B,!
A
Det
erm
ine
the
rela
tion
ship
bet
wee
n t
he
mea
sure
s of
th
e gi
ven
an
gles
.
4.!
R,!
RU
Sm
!R
'm
!R
US
5.!
T,!
US
Tm
!T
%m
!U
ST
6.!
UV
S,!
Rm
!U
VS
%m
!R
Det
erm
ine
the
rela
tion
ship
bet
wee
n t
he
len
gth
s of
th
e gi
ven
sid
es.
7.A !
C!,B!
C!A
C%
BC
8.B!
C!,D!
B!B
C%
DB
9.A!
C!,D!
B!A
C%
DB
ABC
D30
$
30$ 30
$
90$
RV
S
UT
2513
2424
22
21.6
35
AC
B
3.8
4.3
4.0
RT
S
60$
80$
40$
TS
R48
cm
23.7
cm
35 c
m
Answers (Lesson 5-2)
© Glencoe/McGraw-Hill A6 Glencoe Geometry
Skil
ls P
ract
ice
Ineq
ualit
ies
and
Tria
ngle
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
5-2
5-2
©G
lenc
oe/M
cGra
w-H
ill25
3G
lenc
oe G
eom
etry
Lesson 5-2
Det
erm
ine
wh
ich
an
gle
has
th
e gr
eate
st m
easu
re.
1.!
1,!
3,!
42.
!4,
!5,
!7
!1
!4
3.!
2,!
3,!
64.
!5,
!6,
!8
!6
!8
Use
th
e E
xter
ior
An
gle
Ineq
ual
ity
Th
eore
m t
o li
st a
ll
angl
es t
hat
sat
isfy
th
e st
ated
con
dit
ion
.
5.al
l ang
les
who
se m
easu
res
are
less
tha
n m
!1
!2,
!3,
!4,
!5,
!7,
!8
6.al
l ang
les
who
se m
easu
res
are
less
tha
n m
!9
!2,
!4,
!6,
!7
7.al
l ang
les
who
se m
easu
res
are
grea
ter
than
m!
5!
1,!
3
8.al
l ang
les
who
se m
easu
res
are
grea
ter
than
m!
8!
1,!
3,!
5
Det
erm
ine
the
rela
tion
ship
bet
wee
n t
he
mea
sure
s of
th
e gi
ven
an
gles
.
9.m
!A
BD
,m!
BA
D10
.m!
AD
B,m
!B
AD
m!
AB
D%
m!
BA
Dm
!A
DB
'm
!B
AD
11.m
!B
CD
,m!
CD
B12
.m!
CB
D,m
!C
DB
m!
BC
D%
m!
CD
Bm
!C
BD
%m
!C
DB
Det
erm
ine
the
rela
tion
ship
bet
wee
n t
he
len
gth
s of
th
e gi
ven
sid
es.
13.L !
M!,L!
P!14
.M!P!,
M!N!
LM'
LPM
P%
MN
15.M!
N!,N!
P!16
.M!P!,
L!P!M
N'
NP
MP
'LP
83$
57$
79$
44$
59$
38$
LN
P
M
2334
4139
35A
BC
D
1
24
6
7
89
35
12
46
78
35
©G
lenc
oe/M
cGra
w-H
ill25
4G
lenc
oe G
eom
etry
Det
erm
ine
wh
ich
an
gle
has
th
e gr
eate
st m
easu
re.
1.!
1,!
3,!
42.
!4,
!8,
!9
!1
!4
3.!
2,!
3,!
74.
!7,
!8,
!10
!7
!10
Use
th
e E
xter
ior
An
gle
Ineq
ual
ity
Th
eore
m t
o li
st
all
angl
es t
hat
sat
isfy
th
e st
ated
con
dit
ion
.
5.al
l ang
les
who
se m
easu
res
are
less
tha
n m
!1
!3,
!4,
!5,
!7,
!8
6.al
l ang
les
who
se m
easu
res
are
less
tha
n m
!3
!5,
!7,
!8
7.al
l ang
les
who
se m
easu
res
are
grea
ter
than
m!
7!
1,!
3,!
5,!
9
8.al
l ang
les
who
se m
easu
res
are
grea
ter
than
m!
2!
6,!
9
Det
erm
ine
the
rela
tion
ship
bet
wee
n t
he
mea
sure
s of
th
e gi
ven
an
gles
.
9.m
!Q
RW
,m!
RW
Q10
.m!
RT
W,m
!T
WR
m!
QR
W'
!R
WQ
m!
RTW
'!
TWR
11.m
!R
ST
,m!
TR
S12
.m!
WQ
R,m
!Q
RW
m!
RS
T%
!TR
Sm
!W
QR
'!
QR
W
Det
erm
ine
the
rela
tion
ship
bet
wee
n t
he
len
gth
s of
th
e gi
ven
sid
es.
13.D !
H!,G!
H!14
.D!E!
,D!G!
DH
%G
HD
E'
DG
15.E!
G!,F!
G!16
.D!E!
,E!G!
EG
'FG
DE
%E
G
17.S
PORT
ST
he f
igur
e sh
ows
the
posi
tion
of
thre
e tr
ees
on o
ne
part
of
a F
risb
ee™
cou
rse.
At
whi
ch t
ree
posi
tion
is t
he a
ngle
be
twee
n th
e tr
ees
the
grea
test
?2
53 ft
40 ft
3
2
1
37.5
ft
120$
32$
48$
113$
17$
H
DE
F
G
3447
45
44
22
1435
Q
R
S
TW
12
46
78
9
35
12
46
78
910
3
5
Pra
ctic
e (A
vera
ge)
Ineq
ualit
ies
and
Tria
ngle
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
5-2
5-2
Answers (Lesson 5-2)
© Glencoe/McGraw-Hill A7 Glencoe Geometry
An
swer
s
Rea
din
g t
o L
earn
Math
emati
csIn
equa
litie
s an
d Tr
iang
les
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
5-2
5-2
©G
lenc
oe/M
cGra
w-H
ill25
5G
lenc
oe G
eom
etry
Lesson 5-2
Pre-
Act
ivit
yH
ow c
an y
ou t
ell
wh
ich
cor
ner
is
bigg
er?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 5-
2 at
the
top
of
page
247
in y
our
text
book
.
•W
hich
sid
e of
the
pat
io is
opp
osit
e th
e la
rges
t co
rner
?th
e 51
-foo
t si
de•
Whi
ch s
ide
of t
he p
atio
is o
ppos
ite
the
smal
lest
cor
ner?
the
45-fo
ot s
ide
Rea
din
g t
he
Less
on
1.N
ame
the
prop
erty
of
ineq
ualit
y th
at is
illu
stra
ted
by e
ach
of t
he f
ollo
win
g.a.
If x
&8
and
8 &
y,th
en x
&y.
Tran
sitiv
e P
rope
rty
b.If
x%
y,th
en x
#7.
5 %
y#
7.5.
Sub
trac
tion
Pro
pert
yc.
If x
&y,
then
#3x
%#
3y.
Mul
tiplic
atio
n P
rope
rty
d.If
xis
any
rea
l num
ber,
x&
0,x
!0,
or x
%0.
Com
pari
son
Pro
pert
y
2.U
se t
he d
efin
itio
n of
ineq
ualit
y to
wri
te a
n eq
uati
onth
at s
how
s th
at e
ach
ineq
ualit
y is
tru
e.a.
20 &
1220
#12
!8
b.10
1 &
9910
1 #
99 !
2c.
8 &
#2
8 #
"2
!10
d.7
&#
77
#"
7 !
14e.
#11
&#
12"
11 #
"12
!1
f.#
30 &
#45
"30
#"
45 !
15
3.In
the
fig
ure,
m!
IJK
!45
and
m!
H&
m!
I.a.
Arr
ange
the
fol
low
ing
angl
es in
ord
er f
rom
larg
est
to
smal
lest
:!I,
!IJ
K,!
H,!
IJH
!IJ
H,!
IJK
,!H
,!I
b.A
rran
ge t
he s
ides
of "
HIJ
in o
rder
fro
m s
hort
est
to lo
nges
t.H!
J!,I!J!
,H!I!
c.Is
"H
IJan
acu
te,r
ight
,or
obtu
se t
rian
gle?
Exp
lain
you
r re
ason
ing.
Obt
use;
sam
ple
answ
er:!
IJH
is o
btus
e be
caus
e m
!IJ
H#
180
"m
!IJ
K#
135.
Ther
efor
e,"
HIJ
is o
btus
e be
caus
e it
has
an o
btus
e an
gle.
d.Is
"H
IJsc
alen
e,is
osce
les,
or e
quila
tera
l? E
xpla
in y
our
reas
onin
g.S
cale
ne;s
ampl
e an
swer
:the
thr
ee a
ngle
s of
"H
IJal
l hav
e di
ffer
ent
mea
sure
s,so
the
sid
es o
ppos
ite t
hem
mus
t ha
ve d
iffer
ent
leng
ths.
Hel
pin
g Y
ou
Rem
emb
er4.
A g
ood
way
to
rem
embe
r a
new
geo
met
ric
theo
rem
is t
o re
late
it t
o a
theo
rem
you
lear
ned
earl
ier.
Exp
lain
how
the
Ext
erio
r A
ngle
Ine
qual
ity
The
orem
is r
elat
ed t
o th
eE
xter
ior
Ang
le T
heor
em,a
nd w
hy t
he E
xter
ior
Ang
le I
nequ
alit
y T
heor
em m
ust
be t
rue
ifth
e E
xter
ior
Ang
le T
heor
em is
tru
e.S
ampl
e an
swer
:The
Ext
erio
r A
ngle
The
orem
say
s th
at t
he m
easu
re o
f an
exte
rior
ang
le o
f a
tria
ngle
is e
qual
to
the
sum
of
the
mea
sure
s of
the
two
rem
ote
inte
rior
ang
les,
whi
le t
he E
xter
ior
Ang
le In
equa
lity
Theo
rem
says
tha
t th
e m
easu
re o
f an
ext
erio
r an
gle
is g
reat
er t
han
the
mea
sure
of
eith
er r
emot
e in
teri
or a
ngle
.If
a nu
mbe
r is
equ
al t
o th
e su
m o
f tw
opo
sitiv
e nu
mbe
rs,i
t m
ust
be g
reat
er t
han
each
of
thos
e tw
o nu
mbe
rs.
KJ
H
I
©G
lenc
oe/M
cGra
w-H
ill25
6G
lenc
oe G
eom
etry
Con
stru
ctio
n P
robl
emT
he d
iagr
am b
elow
sho
ws
segm
ent
AB
adja
cent
to
a cl
osed
reg
ion.
The
prob
lem
req
uire
s th
at y
ou c
onst
ruct
ano
ther
seg
men
t X
Yto
the
rig
ht o
f th
ecl
osed
reg
ion
such
tha
t po
ints
A,B
,X,a
nd Y
are
colli
near
.You
are
not
allo
wed
to t
ouch
or
cros
s th
e cl
osed
reg
ion
wit
h yo
ur c
ompa
ss o
r st
raig
hted
ge.
Fol
low
th
ese
inst
ruct
ion
s to
con
stru
ct a
seg
men
t X
Yso
th
at i
t is
coll
inea
r w
ith
seg
men
t A
B.
1.C
onst
ruct
the
per
pend
icul
ar b
isec
tor
of A !
B!.L
abel
the
mid
poin
t as
poi
nt C
,an
d th
e lin
e as
m.
2.M
ark
two
poin
ts P
and
Qon
line
mth
at li
e w
ell a
bove
the
clo
sed
regi
on.
Con
stru
ct t
he p
erpe
ndic
ular
bis
ecto
r n
of P!
Q!.L
abel
the
inte
rsec
tion
of
lines
man
d n
as p
oint
D.
3.M
ark
poin
ts R
and
Son
line
nth
at li
e w
ell t
o th
e ri
ght
of t
he c
lose
dre
gion
.Con
stru
ct t
he p
erpe
ndic
ular
bis
ecto
r k
of R!
S!.L
abel
the
inte
rsec
tion
of
lines
nan
d k
as p
oint
E.
4.M
ark
poin
t X
on li
ne k
so t
hat
Xis
bel
ow li
ne n
and
so t
hat
E!X!
isco
ngru
ent
to D !
C!.
5.M
ark
poin
ts T
and
Von
line
kan
d on
opp
osit
e si
des
of X
,so
that
X!T!
and
X !V!
are
cong
ruen
t.C
onst
ruct
the
per
pend
icul
ar b
isec
tor
!of
T!V!
.Cal
l the
poin
t w
here
the
line
!hi
ts t
he b
ound
ary
of t
he c
lose
d re
gion
poi
nt Y
.X !Y!
corr
espo
nds
to t
he n
ew r
oad.
Q Pm
k
!
nD
RE T X V
YB
AC
S
Exis
ting
Road
Clos
ed R
egio
n(L
ake)
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
5-2
5-2
Answers (Lesson 5-2)
© Glencoe/McGraw-Hill A8 Glencoe Geometry
Stu
dy
Gu
ide
and I
nte
rven
tion
Indi
rect
Pro
of
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
5-3
5-3
©G
lenc
oe/M
cGra
w-H
ill25
7G
lenc
oe G
eom
etry
Lesson 5-3
Ind
irec
t Pr
oo
f w
ith
Alg
ebra
One
way
to
prov
e th
at a
sta
tem
ent
is t
rue
is t
o as
sum
eth
at it
s co
nclu
sion
is f
alse
and
the
n sh
ow t
hat
this
ass
umpt
ion
lead
s to
a c
ontr
adic
tion
of
the
hypo
thes
is,a
def
init
ion,
post
ulat
e,th
eore
m,o
r ot
her
stat
emen
t th
at is
acc
epte
d as
tru
e.T
hat
cont
radi
ctio
n m
eans
tha
t th
e co
nclu
sion
can
not
be f
alse
,so
the
conc
lusi
on m
ust
betr
ue.T
his
is k
now
n as
in
dir
ect
pro
of.
Ste
ps fo
r Wri
ting
an In
dire
ct P
roof
1.A
ssum
e th
at th
e co
nclu
sion
is fa
lse.
2.S
how
that
this
ass
umpt
ion
lead
s to
a c
ontr
adic
tion.
3.P
oint
out
that
the
assu
mpt
ion
mus
t be
fals
e, a
nd th
eref
ore,
the
conc
lusi
on m
ust b
e tr
ue.
Giv
en:3
x!
5 %
8P
rove
:x%
1St
ep 1
Ass
ume
that
xis
not
gre
ater
tha
n 1.
Tha
t is
,x!
1 or
x%
1.St
ep 2
Mak
e a
tabl
e fo
r se
vera
l pos
sibi
litie
s fo
r x
!1
or x
%1.
The
cont
radi
ctio
n is
tha
t w
hen
x!
1 or
x%
1,th
en 3
x"
5 is
not
grea
ter
than
8.
Step
3T
his
cont
radi
cts
the
give
n in
form
atio
n th
at 3
x"
5 &
8.T
heas
sum
ptio
n th
at x
is n
ot g
reat
er t
han
1 m
ust
be f
alse
,whi
ch
mea
ns t
hat
the
stat
emen
t “x
&1”
mus
t be
tru
e.
Wri
te t
he
assu
mp
tion
you
wou
ld m
ake
to s
tart
an
in
dir
ect
pro
of o
f ea
ch s
tate
men
t.
1.If
2x
&14
,the
n x
&7.
x(
7
2.Fo
r al
l rea
l num
bers
,if a
"b
&c,
then
a&
c#
b.a
(c
"b
Com
ple
te t
he
pro
of.
Giv
en:n
is a
n in
tege
r an
d n2
is e
ven.
Pro
ve:n
is e
ven.
3.A
ssum
e th
at n
is n
ot e
ven.
That
is,a
ssum
e n
is o
dd.
4.T
hen
nca
n be
exp
ress
ed a
s 2a
"1
by th
e m
eani
ng o
f od
d nu
mbe
r.
5.n2
!(2
a!
1)2
Subs
titu
tion
6.!
(2a
!1)
(2a
!1)
Mul
tipl
y.
7.!
4a2
!4a
!1
Sim
plif
y.
8.!
2(2a
2"
2a) "
1D
istr
ibut
ive
Pro
pert
y
9.2(
2a2
"2a
)"1
is a
n od
d nu
mbe
r.T
his
cont
radi
cts
the
give
n th
at n
2is
eve
n,
so t
he a
ssum
ptio
n m
ust
be fa
lse.
10.T
here
fore
,nis
eve
n.
x3x
"5
18
05
#1
2
#2
#1
#3
#4
Exam
ple
Exam
ple
Exercis
esExercis
es
©G
lenc
oe/M
cGra
w-H
ill25
8G
lenc
oe G
eom
etry
Ind
irec
t Pr
oo
f w
ith
Geo
met
ryTo
wri
te a
n in
dire
ct p
roof
in g
eom
etry
,you
ass
ume
that
the
con
clus
ion
is f
alse
.The
n yo
u sh
ow t
hat
the
assu
mpt
ion
lead
s to
a c
ontr
adic
tion
.T
he c
ontr
adic
tion
sho
ws
that
the
con
clus
ion
cann
ot b
e fa
lse,
so it
mus
t be
tru
e.
Giv
en:m
!C
#10
0P
rove
:!A
is n
ot a
rig
ht
angl
e.St
ep 1
Ass
ume
that
!A
is a
rig
ht a
ngle
.
Step
2Sh
ow t
hat
this
lead
s to
a c
ontr
adic
tion
.If !
Ais
a r
ight
ang
le,
then
m!
A!
90 a
nd m
!C
"m
!A
!10
0 "
90 !
190.
Thu
s th
e su
m o
f th
e m
easu
res
of t
he a
ngle
s of
"A
BC
is g
reat
er t
han
180.
Step
3T
he c
oncl
usio
n th
at t
he s
um o
f th
e m
easu
res
of t
he a
ngle
s of
"
AB
Cis
gre
ater
tha
n 18
0 is
a c
ontr
adic
tion
of
a kn
own
prop
erty
.T
he a
ssum
ptio
n th
at !
Ais
a r
ight
ang
le m
ust
be f
alse
,whi
ch
mea
ns t
hat
the
stat
emen
t “!
Ais
not
a r
ight
ang
le”
mus
t be
tru
e.
Wri
te t
he
assu
mp
tion
you
wou
ld m
ake
to s
tart
an
in
dir
ect
pro
of o
f ea
chst
atem
ent.
1.If
m!
A!
90,t
hen
m!
B!
45.
m!
B)
45
2.If
A!V!
is n
ot c
ongr
uent
to
V!E!
,the
n "
AVE
is n
ot is
osce
les.
"A
VE
is is
osce
les.
Com
ple
te t
he
pro
of.
Giv
en:!
1 "
!2
and
D!G!
is n
ot c
ongr
uent
to
F!G!.
Pro
ve:D !
E!is
not
con
grue
nt t
o F!E!
.
3.A
ssum
e th
at D!
E!$
F!E!.
Ass
ume
the
conc
lusi
on is
fal
se.
4.E!
G!"
E!G!
Ref
lexi
ve P
rope
rty
5."
ED
G"
"E
FG
SA
S
6.D!
G!$
F!G!C
PC
TC
7.T
his
cont
radi
cts
the
give
n in
form
atio
n,so
the
ass
umpt
ion
mus
t
be fa
lse.
8.T
here
fore
,D!
E!is
not
con
grue
nt t
o F!E!
.
12
DG
FE
AB
C
Stu
dy
Gu
ide
and I
nte
rven
tion
(con
tinu
ed)
Indi
rect
Pro
of
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
5-3
5-3
Exercis
esExercis
es
Exam
ple
Exam
ple
Answers (Lesson 5-3)
© Glencoe/McGraw-Hill A9 Glencoe Geometry
An
swer
s
Skil
ls P
ract
ice
Indi
rect
Pro
of
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
5-3
5-3
©G
lenc
oe/M
cGra
w-H
ill25
9G
lenc
oe G
eom
etry
Lesson 5-3
Wri
te t
he
assu
mp
tion
you
wou
ld m
ake
to s
tart
an
in
dir
ect
pro
of o
f ea
ch s
tate
men
t.
1.m
!A
BC
%m
!C
BA
m!
AB
C*
m!
CB
A
2."
DE
F"
"R
ST
"D
EF
%"
RS
T
3.L
ine
ais
per
pend
icul
ar t
o lin
e b.
Line
ais
not
per
pend
icul
ar t
o lin
e b.
4.!
5 is
sup
plem
enta
ry t
o !
6.
!5
is n
ot s
uppl
emen
tary
to
!6.
PRO
OF
Wri
te a
n i
nd
irec
t p
roof
.
5.G
iven
:x2
"8
'12
Pro
ve:x
'2
Pro
of:
Ste
p 1:
Ass
ume
x%
2.S
tep
2:If
x%
2,th
en x
2%
4.B
ut if
x2
%4,
it fo
llow
s th
at x
2!
8 %
12.
This
con
trad
icts
the
giv
en f
act
that
x2
!8
(12
.S
tep
3:S
ince
the
ass
umpt
ion
of x
%2
lead
s to
a c
ontr
adic
tion,
it m
ust
be f
alse
.The
refo
re, x
(2
mus
t be
tru
e.
6.G
iven
:!D
#!
F.
Pro
ve:D
E(
EF
Pro
of:
Ste
p 1:
Ass
ume
DE
#E
F.S
tep
2:If
DE
#E
F,th
en D!
E!$
E!F!by
the
defin
ition
of c
ongr
uent
seg
men
ts.
But
if D!
E!$
E!F!,t
hen
!D
$!
Fby
the
Isos
cele
s Tr
iang
le T
heor
em.
This
con
trad
icts
the
giv
en in
form
atio
n th
at !
D%
!F.
Ste
p 3:
Sin
ce t
he a
ssum
ptio
n th
at D
E#
EF
lead
s to
a c
ontr
adic
tion,
itm
ust
be f
alse
.The
refo
re,i
t m
ust
be t
rue
that
DE
)E
F.
DF
E
©G
lenc
oe/M
cGra
w-H
ill26
0G
lenc
oe G
eom
etry
Wri
te t
he
assu
mp
tion
you
wou
ld m
ake
to s
tart
an
in
dir
ect
pro
of o
f ea
ch s
tate
men
t.
1.B!
D!bi
sect
s !
AB
C.
B!D!
does
not
bis
ect
!A
BC
.
2.R
T!
TS
RT
)TS
PRO
OF
Wri
te a
n i
nd
irec
t p
roof
.
3.G
iven
:#4x
"2
%#
10P
rove
:x&
3P
roof
:S
tep
1:A
ssum
e x
(3.
Ste
p 2:
If x
(3,
then
"4x
*"
12.B
ut "
4x*
"12
impl
ies
that
"
4x!
2 *
"10
,whi
ch c
ontr
adic
ts t
he g
iven
ineq
ualit
y.S
tep
3:S
ince
the
ass
umpt
ion
that
x(
3 le
ads
to a
con
trad
ictio
n,it
mus
t be
tru
e th
at x
%3.
4.G
iven
:m!
2 "
m!
3 (
180
Pro
ve: a
⁄|| bP
roof
:S
tep
1:A
ssum
e a
|| b.
Ste
p 2:
If a
|| b,t
hen
the
cons
ecut
ive
inte
rior
ang
les
!2
and
!3
are
supp
lem
enta
ry.T
hus
m!
2 !
m!
3 #
180.
This
con
trad
icts
the
give
n st
atem
ent
that
m!
2 !
m!
3 )
180.
Ste
p 3:
Sin
ce t
he a
ssum
ptio
n le
ads
to a
con
trad
ictio
n,th
e st
atem
ent
a|| b
mus
t be
fal
se.T
here
fore
,a⁄|| b
mus
t be
tru
e.
5.PH
YSI
CS
Soun
d tr
avel
s th
roug
h ai
r at
abo
ut 3
44 m
eter
s pe
r se
cond
whe
n th
ete
mpe
ratu
re is
20°
C.I
f E
nriq
ue li
ves
2 ki
lom
eter
s fr
om t
he f
ire
stat
ion
and
it t
akes
5
seco
nds
for
the
soun
d of
the
fir
e st
atio
n si
ren
to r
each
him
,how
can
you
pro
vein
dire
ctly
tha
t it
is n
ot 2
0°C
whe
n E
nriq
ue h
ears
the
sir
en?
Ass
ume
that
it is
20°
C w
hen
Enr
ique
hea
rs t
he s
iren
,the
n sh
ow t
hat
atth
is t
empe
ratu
re it
will
tak
e m
ore
than
5 s
econ
ds fo
r th
e so
und
of t
hesi
ren
to r
each
him
.Sin
ce t
he a
ssum
ptio
n is
fal
se,y
ou w
ill h
ave
prov
edth
at it
is n
ot 2
0°C
whe
n E
nriq
ue h
ears
the
sir
en.
1 23
a b
Pra
ctic
e (A
vera
ge)
Indi
rect
Pro
of
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
5-3
5-3
Answers (Lesson 5-3)
© Glencoe/McGraw-Hill A10 Glencoe Geometry
Rea
din
g t
o L
earn
Math
emati
csIn
dire
ct P
roof
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
5-3
5-3
©G
lenc
oe/M
cGra
w-H
ill26
1G
lenc
oe G
eom
etry
Lesson 5-3
Pre-
Act
ivit
yH
ow i
s in
dir
ect
pro
of u
sed
in
lit
erat
ure
?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 5-
3 at
the
top
of
page
255
in y
our
text
book
.
How
cou
ld t
he a
utho
r of
a m
urde
r m
yste
ry u
se in
dire
ct r
easo
ning
to
show
that
a p
arti
cula
r su
spec
t is
not
gui
lty?
Sam
ple
answ
er:A
ssum
e th
atth
e pe
rson
is g
uilty
.The
n sh
ow th
at th
is a
ssum
ptio
n co
ntra
dict
sev
iden
ce t
hat
has
been
gat
here
d ab
out
the
crim
e.
Rea
din
g t
he
Less
on
1.Su
pply
the
mis
sing
wor
ds t
o co
mpl
ete
the
list
of s
teps
invo
lved
in w
riti
ng a
n in
dire
ct p
roof
.
Ste
p 1
Ass
ume
that
the
con
clus
ion
is
.
Ste
p 2
Show
tha
t th
is a
ssum
ptio
n le
ads
to a
of
the
or s
ome
othe
r fa
ct,s
uch
as a
def
init
ion,
post
ulat
e,
,or
coro
llary
.
Ste
p 3
Poin
t ou
t th
at t
he a
ssum
ptio
n m
ust
be
and,
ther
efor
e,th
e
conc
lusi
on m
ust
be
.
2.St
ate
the
assu
mpt
ion
that
you
wou
ld m
ake
to s
tart
an
indi
rect
pro
of o
f ea
ch s
tate
men
t.
a.If
#6x
&30
,the
n x
%#
5.x
*"
5b.
If n
is a
mul
tipl
e of
6,t
hen
nis
a m
ulti
ple
of 3
.n
is n
ot a
mul
tiple
of
3.c.
If a
and
bar
e bo
th o
dd,t
hen
abis
odd
.ab
is e
ven.
abis
gre
ater
d.If
ais
pos
itiv
e an
d b
is n
egat
ive,
then
ab
is n
egat
ive.
than
or
equa
l to
0.e.
If F
is b
etw
een
Ean
d D
,the
n E
F"
FD
!E
D.
EF
!FD
)E
Df.
In a
pla
ne,i
f tw
o lin
es a
re p
erpe
ndic
ular
to
the
sam
e lin
e,th
en t
hey
are
para
llel.
Two
lines
are
not
par
alle
l.g.
Ref
er t
o th
e fi
gure
.h
.R
efer
to
the
figu
re.
If A
B!
AC
,the
n m
!B
!m
!C
.In
"P
QR
,PR
"Q
R&
QP
.m
!B
)m
!C
PR
!Q
R(
QP
Hel
pin
g Y
ou
Rem
emb
er3.
A g
ood
way
to
rem
embe
r a
new
con
cept
in m
athe
mat
ics
is t
o re
late
it t
o so
met
hing
you
hav
eal
read
y le
arne
d.H
ow is
the
pro
cess
of i
ndir
ect
proo
f rel
ated
to
the
rela
tion
ship
bet
wee
n a
cond
itio
nal s
tate
men
t an
d it
s co
ntra
posi
tive
?S
ampl
e an
swer
:The
con
trap
ositi
veof
the
con
ditio
nal s
tate
men
t p
→q
is t
he s
tate
men
t &
q→
&p.
In a
nin
dire
ct p
roof
of
a co
nditi
onal
sta
tem
ent
p→
q,yo
u as
sum
e th
at q
isfa
lse
and
show
tha
t th
is im
plie
s th
at p
is f
alse
,tha
t is
,you
sho
w t
hat
&q
→&
pis
tru
e.B
ecau
se a
sta
tem
ent
is lo
gica
lly e
quiv
alen
t to
its
cont
rapo
sitiv
e,pr
ovin
g th
e co
ntra
posi
tive
is t
rue
is a
way
of
prov
ing
the
orig
inal
con
ditio
nal i
s tr
ue.
PRQ
AC
B
true
fals
eth
eore
mhy
poth
esis
cont
radi
ctio
nfa
lse
©G
lenc
oe/M
cGra
w-H
ill26
2G
lenc
oe G
eom
etry
Mor
e C
ount
erex
ampl
esSo
me
stat
emen
ts in
mat
hem
atic
s ca
n be
pro
ven
fals
e by
cou
nte
rexa
mp
les.
Con
side
r th
e fo
llow
ing
stat
emen
t.
For
any
num
bers
aan
d b,
a#
b!
b#
a.
You
can
prov
e th
at t
his
stat
emen
t is
fal
se in
gen
eral
if y
ou c
an f
ind
one
exam
ple
for
whi
ch t
he s
tate
men
t is
fal
se.
Let
a!
7 an
d b
!3.
Subs
titu
te t
hese
val
ues
in t
he e
quat
ion
abov
e.
7 #
3 #
3 #
74
( #
4
In g
ener
al,f
or a
ny n
umbe
rs a
and
b,th
e st
atem
ent
a#
b!
b#
ais
fal
se.
You
can
mak
e th
e eq
uiva
lent
ver
bal s
tate
men
t:su
btra
ctio
n is
not
aco
mm
utat
ive
oper
atio
n.
In e
ach
of
the
foll
owin
g ex
erci
ses
a,b
,an
d c
are
any
nu
mbe
rs.P
rove
th
at
the
stat
emen
t is
fal
se b
y co
un
tere
xam
ple
.S
ampl
e an
swer
s ar
e gi
ven.
1.a
#(b
#c)
# (a
#b)
#c
2.a
)(b
)c)
# (a
)b)
)c
6 "
(4 "
2) #
(6 "
4) "
26
+(4
+2)
# (6
+4)
+2
6 "
2 #
2 "
2&6 2&
#&1 2.5 &
4 )
03
) 0
.75
3.a
)b
# b
)a
4.a
)(b
"c)
# (a
)b)
"(a
)c)
6 +
4 #
4 +
66
+(4
!2)
#(6
+4)
!(6
+2)
&3 2&)
&2 3&6
+6
#1.
5 !
31
) 4
.5
5.a
"(b
c) #
(a"
b)(a
"c)
6.a2
"a2
# a
4
6 !
(4 ,
2)
#(6
!4)
(6 !
2)62
!62
#64
6 !
8 #
(10)
(8)
36 !
36 #
1296
14 )
80
72 )
129
6
7.W
rite
the
ver
bal e
quiv
alen
ts f
or E
xerc
ises
1,2
,and
3.
1.S
ubtr
actio
n is
not
an
asso
ciat
ive
oper
atio
n.2.
Div
isio
n is
not
an
asso
ciat
ive
oper
atio
n.3.
Div
isio
n is
not
a c
omm
utat
ive
oper
atio
n.
8.Fo
r th
e D
istr
ibut
ive
Pro
pert
y a(
b"
c) !
ab"
acit
is s
aid
that
mul
tipl
icat
ion
dist
ribu
tes
over
add
itio
n.E
xerc
ises
4 a
nd 5
pro
ve t
hat
som
e op
erat
ions
do
not
dist
ribu
te.W
rite
a s
tate
men
t fo
r ea
ch e
xerc
ise
that
indi
cate
s th
is.
4.D
ivis
ion
does
not
dis
trib
ute
over
add
ition
.5.
Add
ition
doe
s no
t di
stri
bute
ove
r m
ultip
licat
ion.
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
5-3
5-3
Answers (Lesson 5-3)
© Glencoe/McGraw-Hill A11 Glencoe Geometry
An
swer
s
Stu
dy
Gu
ide
and I
nte
rven
tion
The
Tria
ngle
Ineq
ualit
y
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
5-4
5-4
©G
lenc
oe/M
cGra
w-H
ill26
3G
lenc
oe G
eom
etry
Lesson 5-4
The
Tria
ng
le In
equ
alit
yIf
you
tak
e th
ree
stra
ws
of le
ngth
s 8
inch
es,5
inch
es,a
nd
1 in
ch a
nd t
ry t
o m
ake
a tr
iang
le w
ith
them
,you
will
fin
d th
at it
is n
ot p
ossi
ble.
Thi
sill
ustr
ates
the
Tri
angl
e In
equa
lity
The
orem
.
Tria
ngle
Ineq
ualit
yT
he s
um o
f the
leng
ths
of a
ny tw
o si
des
of a
Theo
rem
tria
ngle
is g
reat
er th
an th
e le
ngth
of t
he th
ird s
ide.
Th
e m
easu
res
of t
wo
sid
es o
f a
tria
ngl
e ar
e 5
and
8.F
ind
a r
ange
for
the
len
gth
of
the
thir
d s
ide.
By
the
Tri
angl
e In
equa
lity,
all t
hree
of
the
follo
win
g in
equa
litie
s m
ust
be t
rue.
5 "
x&
88
"x
&5
5 "
8 &
xx
&3
x&
#3
13 &
x
The
refo
re x
mus
t be
bet
wee
n 3
and
13.
Det
erm
ine
wh
eth
er t
he
give
n m
easu
res
can
be
the
len
gth
s of
th
e si
des
of
atr
ian
gle.
Wri
te y
esor
no.
1.3,
4,6
yes
2.6,
9,15
no
3.8,
8,8
yes
4.2,
4,5
yes
5.4,
8,16
no6.
1.5,
2.5,
3ye
s
Fin
d t
he
ran
ge f
or t
he
mea
sure
of
the
thir
d s
ide
give
n t
he
mea
sure
s of
tw
o si
des
.
7.1
and
6 8.
12 a
nd 1
8
5 '
n'
76
'n
'30
9.1.
5 an
d 5.
5 10
.82
and
8
4 '
n'
774
'n
'90
11.S
uppo
se y
ou h
ave
thre
e di
ffer
ent
posi
tive
num
bers
arr
ange
d in
ord
er f
rom
leas
t to
grea
test
.Wha
t si
ngle
com
pari
son
will
let
you
see
if t
he n
umbe
rs c
an b
e th
e le
ngth
s of
the
side
s of
a t
rian
gle?
Find
the
sum
of
the
two
smal
ler
num
bers
.If
that
sum
is g
reat
er t
han
the
larg
est
num
ber,
then
the
thr
ee n
umbe
rs c
an b
e th
e le
ngth
s of
the
sid
esof
a t
rian
gle.
BC
A
a
cb
Exercis
esExercis
es
Exam
ple
Exam
ple
©G
lenc
oe/M
cGra
w-H
ill26
4G
lenc
oe G
eom
etry
Dis
tan
ce B
etw
een
a P
oin
t an
d a
Lin
e
Stu
dy
Gu
ide
and I
nte
rven
tion
(con
tinu
ed)
The
Tria
ngle
Ineq
ualit
y
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
5-4
5-4
The
per
pend
icul
ar s
egm
ent
from
a p
oint
to
a lin
e is
the
sho
rtes
t se
gmen
t fr
om t
hepo
int
to t
he li
ne.
P!C!is
the
shor
test
seg
men
t fro
m P
toA
B#!
" .
The
per
pend
icul
ar s
egm
ent
from
a p
oint
to
a pl
ane
is t
he s
hort
est
segm
ent
from
the
poin
t to
the
pla
ne.
Q!T!
is th
e sh
orte
st s
egm
ent f
rom
Qto
pla
ne N
.
Q TN
B
P CA G
iven
:Poi
nt
Pis
equ
idis
tan
t fr
om t
he
sid
es
of a
n a
ngl
e.P
rove
:B !A!
$C!
A!P
roof
:1.
Dra
w B !
P!an
d C!
P!⊥
to
1.D
ist.
is m
easu
red
the
side
s of
!R
AS
.al
ong
a ⊥.
2.!
PB
Aan
d !
PC
Aar
e ri
ght
angl
es.
2.D
ef.o
f ⊥
lines
3."
AB
Pan
d "
AC
Par
e ri
ght
tria
ngle
s.3.
Def
.of
rt."
4.!
PB
A"
!P
CA
4.R
t.an
gles
are
".
5.P
is e
quid
ista
nt f
rom
the
sid
es o
f !R
AS
.5.
Giv
en6.
B !P!
"C!
P!6.
Def
.of
equi
dist
ant
7.A !
P!"
A!P!
7.R
efle
xive
Pro
pert
y8.
"A
BP
""
AC
P8.
HL
9.B !
A!"
C!A!
9.C
PC
TC
Com
ple
te t
he
pro
of.
Giv
en:"
AB
C"
"R
ST
;!D
"!
UP
rove
:A !D!
"R!
U!P
roof
:
1."
AB
C"
"R
ST
;!D
"!
U1.
Giv
en2.
A!C!
"R!
T!2.
CP
CTC
3.!
AC
B"
!R
TS
3.C
PC
TC4.
!A
CB
and
!A
CD
are
a lin
ear
pair
;4.
Def
.of lin
ear
pair
!R
TS
and
!R
TU
are
a lin
ear
pair
.
5.!
AC
Ban
d !
AC
Dar
e su
pple
men
tary
;5.
Line
ar p
airs
are
sup
pl.
!R
TS
and
!R
TU
are
supp
lem
enta
ry.
6.!
AC
D$
!R
TU6.
Ang
les
supp
l.to
"an
gles
are
".
7."
AD
C"
"R
UT
7.A
AS
8.A!
D!$
R!U!
8.C
PC
TC
A DC
B
R UT
S
AS
CPB
R
Exam
ple
Exam
ple
Exercis
esExercis
es
Answers (Lesson 5-4)
© Glencoe/McGraw-Hill A12 Glencoe Geometry
Skil
ls P
ract
ice
The
Tria
ngle
Ineq
ualit
y
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
5-4
5-4
©G
lenc
oe/M
cGra
w-H
ill26
5G
lenc
oe G
eom
etry
Lesson 5-4
Det
erm
ine
wh
eth
er t
he
give
n m
easu
res
can
be
the
len
gth
s of
th
e si
des
of
atr
ian
gle.
Wri
te y
esor
no.
1.2,
3,4
yes
2.5,
7,9
yes
3.4,
8,11
yes
4.13
,13,
26no
5.9,
10,2
0no
6.15
,17,
19ye
s
7.14
,17,
31no
8.6,
7,12
yes
Fin
d t
he
ran
ge f
or t
he
mea
sure
of
the
thir
d s
ide
of a
tri
angl
e gi
ven
th
e m
easu
res
of t
wo
sid
es.
9.5
and
910
.7 a
nd 1
4
4 '
n'
147
'n
'21
11.8
and
13
12.1
0 an
d 12
5 '
n'
212
'n
'22
13.1
2 an
d 15
14.1
5 an
d 27
3 '
n'
2712
'n
'42
15.1
7 an
d 28
16.1
8 an
d 22
11 '
n'
454
'n
'40
ALG
EBR
AD
eter
min
e w
het
her
th
e gi
ven
coo
rdin
ates
are
th
e ve
rtic
es o
f a
tria
ngl
e.E
xpla
in.
17.A
(3,5
),B
(4,7
),C
(7,6
)18
.S(6
,5),
T(8
,3),
U(1
2,#
1)
Yes;
AB
#'
5!,B
C#
'10!
,and
N
o;S
T#
2'2!,
TU#
4'2!,
and
AC
#'
17!,s
o A
B!
BC
%A
C,
SU
#6'
2!,so
ST
!TU
#S
U.
AB
!A
C%
BC
,and
A
C!
BC
%A
B.
19.H
(#8,
4),I
(#4,
2),J
(4,#
2)20
.D(1
,#5)
,E(#
3,0)
,F(#
1,0)
No;
HI#
2'5!,
IJ#
4'5!,
and
Yes;
DE
#'
41!,E
F#
2,an
d H
J#
6'5!,
so H
I!IJ
#H
J.D
F#
'29!
,so
DE
!E
F%
DF,
DE
!D
F%
EF,
and
DF
!E
F%
DE
.
©G
lenc
oe/M
cGra
w-H
ill26
6G
lenc
oe G
eom
etry
Det
erm
ine
wh
eth
er t
he
give
n m
easu
res
can
be
the
len
gth
s of
th
e si
des
of
atr
ian
gle.
Wri
te y
esor
no.
1.9,
12,1
8ye
s2.
8,9,
17no
3.14
,14,
19ye
s4.
23,2
6,50
no
5.32
,41,
63ye
s6.
2.7,
3.1,
4.3
yes
7.0.
7,1.
4,2.
1no
8.12
.3,1
3.9,
25.2
yes
Fin
d t
he
ran
ge f
or t
he
mea
sure
of
the
thir
d s
ide
of a
tri
angl
e gi
ven
th
e m
easu
res
of t
wo
sid
es.
9.6
and
1910
.7 a
nd 2
913
'n
'25
22 '
n'
36
11.1
3 an
d 27
12.1
8 an
d 23
14 '
n'
405
'n
'41
13.2
5 an
d 38
14.3
1 an
d 39
13 '
n'
638
'n
'70
15.4
2 an
d 6
16.5
4 an
d 7
36 '
n'
4847
'n
'61
ALG
EBR
AD
eter
min
e w
het
her
th
e gi
ven
coo
rdin
ates
are
th
e ve
rtic
es o
f a
tria
ngl
e.E
xpla
in.
17.R
(1,3
),S
(4,0
),T
(10,
#6)
18.W
(2,6
),X
(1,6
),Y
(4,2
)
No;
RS
#3'
2!,S
T#
6'2!,
and
Yes;
WX
#1,
XY
#5,
and
RT
#9'
2!,so
RS
!S
T#
RT.
WY
#2'
5!,so
WX
!X
Y%
WY
,W
X!
WY
%X
Y,a
nd
WY
!X
Y%
WX
.
19.P
(#3,
2),L
(1,1
),M
(9,#
1)20
.B(1
,1),
C(6
,5),
D(4
,#1)
No;
PL
#'
17!,L
M#
2 '
17!,a
nd
Yes;
BC
#'
41!,C
D#
2'10!
,and
PM
#3
'17!
,so
PL
!LM
#P
M.
BD
#'
13!,s
o B
C!
CD
%B
D,
BC
!B
D%
CD
,and
BD
!C
D%
BC
.
21.G
AR
DEN
ING
Ha
Poon
g ha
s 4
leng
ths
of w
ood
from
whi
ch h
e pl
ans
to m
ake
a bo
rder
for
atr
iang
ular
-sha
ped
herb
gar
den.
The
leng
ths
of t
he w
ood
bord
ers
are
8 in
ches
,10
inch
es,
12 in
ches
,and
18
inch
es.H
ow m
any
diff
eren
t tr
iang
ular
bor
ders
can
Ha
Poon
g m
ake?
3
Pra
ctic
e (A
vera
ge)
The
Tria
ngle
Ineq
ualit
y
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
5-4
5-4
Answers (Lesson 5-4)
© Glencoe/McGraw-Hill A13 Glencoe Geometry
An
swer
s
Rea
din
g t
o L
earn
Math
emati
csTh
e Tr
iang
le In
equa
lity
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
5-4
5-4
©G
lenc
oe/M
cGra
w-H
ill26
7G
lenc
oe G
eom
etry
Lesson 5-4
Pre-
Act
ivit
yH
ow c
an y
ou u
se t
he
Tri
angl
e In
equ
alit
y T
heo
rem
wh
en t
rave
lin
g?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 5-
4 at
the
top
of
page
261
in y
our
text
book
.
In a
ddit
ion
to t
he g
reat
er d
ista
nce
invo
lved
in f
lyin
g fr
om C
hica
go t
oC
olum
bus
thro
ugh
Indi
anap
olis
rat
her
than
fly
ing
nons
top,
wha
t ar
e tw
oot
her
reas
ons
that
it w
ould
tak
e lo
nger
to
get
to C
olum
bus
if y
ou t
ake
two
flig
hts
rath
er t
han
one?
Sam
ple
answ
er:t
ime
need
ed fo
r an
ext
rata
keof
f an
d la
ndin
g;la
yove
r tim
e in
Indi
anap
olis
bet
wee
n th
etw
o fli
ghts
Rea
din
g t
he
Less
on
1.R
efer
to
the
figu
re.
Whi
ch s
tate
men
ts a
re t
rue?
C,D
,FA
.DE
&E
F"
FD
B.D
E!
EF
"F
DC
.EG
!E
F"
FG
D.E
D"
DG
&E
GE
.The
sho
rtes
t di
stan
ce f
rom
Dto
EG
#!" i
s D
F.
F.T
he s
hort
est
dist
ance
fro
m D
to E
G# !
" is
DG
.
2.C
ompl
ete
each
sen
tenc
e ab
out
"X
YZ
.
a.If
XY
!8
and
YZ
!11
,the
n th
e ra
nge
of v
alue
s fo
r X
Zis
%
XZ
%.
b.If
XY
!13
and
XZ
!25
,the
n Y
Zm
ust
be b
etw
een
and
.
c.If
"X
YZ
is is
osce
les
wit
h !
Zas
the
ver
tex
angl
e,an
d X
Z!
8.5,
then
the
ran
ge o
f
valu
es f
or X
Yis
%
XY
%.
d.If
XZ
!a
and
YZ
!b,
wit
h b
%a,
then
the
ran
ge fo
r X
Yis
%
XY
%.
Hel
pin
g Y
ou
Rem
emb
er
3.A
goo
d w
ay t
o re
mem
ber
a ne
w t
heor
em is
to
stat
e it
info
rmal
ly in
dif
fere
nt w
ords
.How
coul
d yo
u re
stat
e th
e T
rian
gle
Ineq
ualit
y T
heor
em?
Sam
ple
answ
er:T
he s
ide
that
con
nect
s on
e ve
rtex
of
a tr
iang
le t
oan
othe
r is
a s
hort
er p
ath
betw
een
the
two
vert
ices
tha
n th
e pa
th t
hat
goes
thr
ough
the
thi
rd v
erte
x.
a!
ba
"b
170
3812
193
ZX
Y
GD
EF
©G
lenc
oe/M
cGra
w-H
ill26
8G
lenc
oe G
eom
etry
Con
stru
ctin
g Tr
iang
les
Th
e m
easu
rem
ents
of
the
sid
es o
f a
tria
ngl
e ar
e gi
ven
.If
a tr
ian
gle
hav
ing
sid
esw
ith
th
ese
mea
sure
men
ts i
s n
ot p
ossi
ble,
then
wri
te i
mpo
ssib
le.I
f a
tria
ngl
e is
pos
sibl
e,d
raw
it
and
mea
sure
eac
h a
ngl
e w
ith
a p
rotr
acto
r.
1.A
R!
5 cm
m!
A!
302.
PI
!8
cmm
!P
!
RT
!3
cmm
!R
!90
IN!
3 cm
m!
I!
AT
!6
cmm
!T
!60
PN
!2
cmm
!N
!
impo
ssib
le
3.O
N!
10 c
mm
!O
!4.
TW
!6
cmm
!T
!11
5
NE
!5.
3 cm
m!
N!
WO
!7
cmm
!W
!15
GE
!4.
6 cm
m!
E!
TO
!2
cmm
!O
!50
impo
ssib
le
5.B
A!
3.l c
mm
!B
!16
36.
AR
!4
cmm
!A
!90
AT
!8
cmm
!A
!11
RM
!5
cmm
!R
!37
BT
!5
cmm
!T
!6
AM
!3
cmm
!M
!53
M
RA
T
BA
W
T
O
AR T
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
5-4
5-4
Answers (Lesson 5-4)
© Glencoe/McGraw-Hill A14 Glencoe Geometry
Stu
dy
Gu
ide
and I
nte
rven
tion
Ineq
ualit
ies
Invo
lvin
g Tw
o Tr
iang
les
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
5-5
5-5
©G
lenc
oe/M
cGra
w-H
ill26
9G
lenc
oe G
eom
etry
Lesson 5-5
SAS
Ineq
ual
ity
The
fol
low
ing
theo
rem
invo
lves
the
rel
atio
nshi
p be
twee
n th
e si
des
oftw
o tr
iang
les
and
an a
ngle
in e
ach
tria
ngle
.
If tw
o si
des
of a
tria
ngle
are
con
grue
nt
to tw
o si
des
of a
noth
er tr
iang
le a
nd th
e in
clud
ed a
ngle
in o
ne tr
iang
le h
as a
S
AS
Ineq
ualit
y/H
inge
The
orem
grea
ter
mea
sure
than
the
incl
uded
ang
le
in th
e ot
her,
then
the
third
sid
e of
the
If
R!S!
"A!B!
, S!T!
"B!C!
, and
first
tria
ngle
is lo
nger
than
the
third
sid
e m
!S
&m
!B
, the
n R
T&
AC
.of
the
seco
nd tr
iang
le.
Wri
te a
n i
neq
ual
ity
rela
tin
g th
e le
ngt
hs
of C !
D!an
d A!
D!.
Tw
o si
des
of "
BC
Dar
e co
ngru
ent
to t
wo
side
s of
"B
AD
and
m!
CB
D&
m!
AB
D.B
y th
e SA
S In
equa
lity/
Hin
ge T
heor
em,
CD
&A
D.
Wri
te a
n i
neq
ual
ity
rela
tin
g th
e gi
ven
pai
r of
seg
men
t m
easu
res.
1.2.
MR
,RP
AD
,CD
MR
%R
PA
D%
CD
3.4.
EG
,HK
MR
,PR
EG
'H
KM
R%
PR
Wri
te a
n i
neq
ual
ity
to d
escr
ibe
the
pos
sibl
e va
lues
of
x.
5.6.
x%
12.5
x%
1.6
62$
65$
2.7
cm1.
8 cm
1.8
cm( 3
x "
2.1
) cm
115$12
0$24
cm
24 c
m40
cm
( 4x
" 1
0) c
m
MR
NP
48$46
$
2025
20
EG
HKJ
F60
$
62$
10
10
42
42
C ADB
22$
38$
N
R
P
M
21$
19$
BD A
28$ 22
$
C
ST
80$
R
BC
60$A
Exam
ple
Exam
ple
Exercis
esExercis
es
©G
lenc
oe/M
cGra
w-H
ill27
0G
lenc
oe G
eom
etry
SSS
Ineq
ual
ity
The
con
vers
e of
the
Hin
ge T
heor
em is
als
o us
eful
whe
n tw
o tr
iang
les
have
tw
o pa
irs
of c
ongr
uent
sid
es.
If tw
o si
des
of a
tria
ngle
are
con
grue
nt to
two
side
sof
ano
ther
tria
ngle
and
the
third
sid
e in
one
tria
ngle
SS
S In
equa
lity
is lo
nger
than
the
third
sid
e in
the
othe
r, th
en th
e an
gle
betw
een
the
pair
of c
ongr
uent
sid
es in
the
first
tria
ngle
is g
reat
er th
an th
e co
rres
pond
ing
angl
e in
the
seco
nd tr
iang
le.
If N
M!
SR
, MP
!R
T, a
nd N
P&
ST,
then
m!
M&
m!
R.
Wri
te a
n i
neq
ual
ity
rela
tin
g th
e m
easu
res
of
!A
BD
and
!C
BD
.T
wo
side
s of
"A
BD
are
cong
ruen
t to
tw
o si
des
of "
CB
D,a
nd A
D&
CD
.B
y th
e SS
S In
equa
lity,
m!
AB
D&
m!
CB
D.
Wri
te a
n i
neq
ual
ity
rela
tin
g th
e gi
ven
pai
r of
an
gle
mea
sure
s.
1.2.
m!
MP
R,m
!N
PR
m!
AB
D,m
!C
BD
m!
MP
R%
m!
NP
Rm
!A
BD
'm
!C
BD
3.4.
m!
C,m
!Z
m!
XY
W,m
!W
YZ
m!
C'
m!
Zm
!X
YW
'm
!W
YZ
Wri
te a
n i
neq
ual
ity
to d
escr
ibe
the
pos
sibl
e va
lues
of
x.
5.6.
12 '
x'
116
1 '
x'
1233$
60 c
m
60 c
m
36 c
m
30 c
m( 3
x "
3) $
(1 – 2x "
6)$
52$
30
30
28
12
42
28
ZW
XY
30C
AX
B30
5048
2424
ZY
1116
2626
B
CD
A
13
10
M
R
NP
13 16C D A
B
3838
2323
3336
TR
SN
MP
Stu
dy
Gu
ide
and I
nte
rven
tion
(con
tinu
ed)
Ineq
ualit
ies
Invo
lvin
g Tw
o Tr
iang
les
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
5-5
5-5
Exam
ple
Exam
ple
Exercis
esExercis
es
Answers (Lesson 5-5)
© Glencoe/McGraw-Hill A15 Glencoe Geometry
An
swer
s
Skil
ls P
ract
ice
Ineq
ualit
ies
Invo
lvin
g Tw
o Tr
iang
les
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
5-5
5-5
©G
lenc
oe/M
cGra
w-H
ill27
1G
lenc
oe G
eom
etry
Lesson 5-5
Wri
te a
n i
neq
ual
ity
rela
tin
g th
e gi
ven
pai
r of
an
gles
or
seg
men
t m
easu
res.
1.m
!B
XA
,m!
DX
Am
!B
XA
'm
!D
XA
2.B
C,D
CB
C%
DC
Wri
te a
n i
neq
ual
ity
rela
tin
g th
e gi
ven
pai
r of
an
gles
or
segm
ent
mea
sure
s.
3.m
!S
TR
,m!
TR
U4.
PQ
,RQ
m!
STR
%m
!TR
UP
Q%
RQ
5.In
the
fig
ure,
B!A!
,B!D!
,B!C!
,and
B!E!
are
cong
ruen
t an
d A
C%
DE
.H
ow d
oes
m!
1 co
mpa
re w
ith
m!
3? E
xpla
in y
our
thin
king
.
m!
1 '
m!
3;Fr
om t
he g
iven
info
rmat
ion
and
the
SS
S In
equa
lity
Theo
rem
,it
follo
ws
that
in "
AB
Can
d "
DB
Ew
e ha
ve m
!A
BC
'm
!D
BE
.Sin
ce
m!
AB
C#
m!
1 !
m!
2 an
d m
!D
BE
#m
!3
!m
!2,
it fo
llow
s th
at m
!1
!m
!2
'm
!3
!m
!2.
Sub
trac
t m
!2
from
eac
h si
de o
f th
e la
st in
equa
lity
to g
et
m!
1 '
m!
3.
6.W
rite
a t
wo-
colu
mn
proo
f.G
iven
:B !A!
"D!
A!B
C&
DC
Pro
ve:m
!1
&m
!2
Pro
of:
Sta
tem
ents
Rea
sons
1.B!
A!$
D!A!
1.G
iven
2.B
C%
DC
2.G
iven
3.A!
C!$
A!C!
3.R
efle
xive
Pro
pert
y4.
m!
1 %
m!
24.
SS
S In
equa
lity
1 2
B
A
D
C
12
3
B
AD
C
E
95$
77
85$
PR
SQ31
30
2222
RS
UT
6
98
3
3
B
AC
D
X
©G
lenc
oe/M
cGra
w-H
ill27
2G
lenc
oe G
eom
etry
Wri
te a
n i
neq
ual
ity
rela
tin
g th
e gi
ven
pai
r of
an
gles
or
segm
ent
mea
sure
s.
1.A
B,B
K2.
ST
,SR
AB
%B
KS
T%
SR
3.m
!C
DF
,m!
ED
F4.
m!
R,m
!T
m!
CD
F'
m!
ED
Fm
!R
'm
!T
5.W
rite
a t
wo-
colu
mn
proo
f.G
iven
:Gis
the
mid
poin
t of
D !F!.
m!
1 &
m!
2P
rove
:ED
&E
F
Pro
of:
Sta
tem
ents
Rea
sons
1.G
is t
he m
idpo
int
of D!
F!.1.
Giv
en2.
D!G!
$F!G!
2.D
efin
ition
of
mid
poin
t3.
E!G!$
E!G!3.
Ref
lexi
ve P
rope
rty
4.m
!1
%m
!2
4.G
iven
5.E
D%
EF
5.S
AS
Ineq
ualit
y
6.TO
OLS
Reb
ecca
use
d a
spri
ng c
lam
p to
hol
d to
geth
er a
cha
ir
leg
she
repa
ired
wit
h w
ood
glue
.Whe
n sh
e op
ened
the
cla
mp,
she
noti
ced
that
the
ang
le b
etw
een
the
hand
les
of t
he c
lam
pde
crea
sed
as t
he d
ista
nce
betw
een
the
hand
les
of t
he c
lam
pde
crea
sed.
At
the
sam
e ti
me,
the
dist
ance
bet
wee
n th
e gr
ippi
ng e
nds
of t
he c
lam
p in
crea
sed.
Whe
n sh
e re
leas
ed t
he
hand
les,
the
dist
ance
bet
wee
n th
e gr
ippi
ng e
nd o
f th
e cl
amp
decr
ease
d an
d th
e di
stan
ce b
etw
een
the
hand
les
incr
ease
d.Is
the
cla
mp
an e
xam
ple
of t
he S
AS
or S
SS I
nequ
alit
y?S
AS
Ineq
ualit
y
12
DF
E G
2021
RT
S
JK
1414
14
13
12C
F
ED
( x !
3) $
( x "
3) $
1010
RT
S
Q
40$
30$
60$
AK
M
B
Pra
ctic
e (A
vera
ge)
Ineq
ualit
ies
Invo
lvin
g Tw
o Tr
iang
les
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
5-5
5-5
Answers (Lesson 5-5)
© Glencoe/McGraw-Hill A16 Glencoe Geometry
Rea
din
g t
o L
earn
Math
emati
csIn
equa
litie
s In
volv
ing
Two
Tria
ngle
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
5-5
5-5
©G
lenc
oe/M
cGra
w-H
ill27
3G
lenc
oe G
eom
etry
Lesson 5-5
Pre-
Act
ivit
yH
ow d
oes
a ba
ckh
oe w
ork
?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 5-
5 at
the
top
of
page
267
in y
our
text
book
.
Wha
t is
the
mai
n ki
nd o
f ta
sk t
hat
back
hoes
are
use
d to
per
form
?B
ackh
oes
are
used
mai
nly
for
digg
ing.
Rea
din
g t
he
Less
on
1.R
efer
to
the
figu
re.W
rite
a c
oncl
usio
n th
at y
ou c
an d
raw
fro
m t
he g
iven
info
rmat
ion.
The
n na
me
the
theo
rem
tha
t ju
stif
ies
your
con
clus
ion.
a.L !M!
"O!
P!,M!
N!"
P!Q!,a
nd L
N&
OQ
m!
M%
m!
P;S
SS
Ineq
ualit
y Th
eore
mb.
L!M!"
O!P!,
M!N!
"P!Q!
,and
m!
P%
m!
MO
Q'
LN(o
r LN
%O
Q);
SA
S In
equa
lity
Theo
rem
(or
Hin
ge T
heor
em)
c.L
M!
8,L
N!
15,O
P!
8,O
Q!
15,m
!L
!22
,and
m!
O!
21M
N%
PQ
;SA
S In
equa
lity
Theo
rem
(or
Hin
ge T
heor
em)
2.In
the
fig
ure,
"E
FG
is is
osce
les
wit
h ba
se F!
G!an
d F
is t
he
mid
poin
t of
D !G!
.Det
erm
ine
whe
ther
eac
h of
the
fol
low
ing
is
a va
lid c
oncl
usio
n th
at y
ou c
an d
raw
bas
ed o
n th
e gi
ven
info
rmat
ion.
(Wri
te v
alid
or i
nval
id.)
If t
he c
oncl
usio
n is
val
id,
iden
tify
the
def
init
ion,
prop
erty
,pos
tula
te,o
r th
eore
m t
hat
supp
orts
it.
a.!
3 "
!4
valid
;Iso
scel
es T
rian
gle
Theo
rem
b.D
F!
GF
valid
;def
initi
on o
f m
idpo
int
c."
DE
Fis
isos
cele
s.in
valid
d.m
!3
&m
!1
valid
;Ext
erio
r A
ngle
Ineq
ualit
y Th
eore
me.
m!
2 &
m!
4va
lid;E
xter
ior
Ang
le In
equa
lity
Theo
rem
f.m
!2
&m
!3
valid
;Sub
stitu
tion
Pro
pert
y (u
sing
con
clus
ions
fro
m p
arts
g.
DE
&E
Gva
lid;S
AS
Ineq
ualit
y Th
eore
m (
or H
inge
The
orem
)
a a
nd e
)
h.
DE
&F
Gin
valid
Hel
pin
g Y
ou
Rem
emb
er3.
A g
ood
way
to
rem
embe
r so
met
hing
is t
o th
ink
of it
in c
oncr
ete
term
s.H
ow c
an y
ouill
ustr
ate
the
Hin
ge T
heor
em w
ith
ever
yday
obj
ects
?S
ampl
e an
swer
:Put
tw
ope
ncils
on
a de
skto
p so
tha
t th
e er
aser
s to
uch.
As
you
incr
ease
or
decr
ease
the
mea
sure
of
the
angl
e fo
rmed
by
the
penc
ils,t
he d
ista
nce
betw
een
the
poin
ts o
f th
e pe
ncils
incr
ease
s or
dec
reas
es a
ccor
ding
ly.
FG
D
E
12
34
NQ
PM
LO
©G
lenc
oe/M
cGra
w-H
ill27
4G
lenc
oe G
eom
etry
Dra
win
g a
Dia
gram
It is
use
ful a
nd o
ften
nec
essa
ry t
o dr
aw a
dia
gram
of
the
situ
atio
nbe
ing
desc
ribe
d in
a p
robl
em.T
he v
isua
lizat
ion
of t
he p
robl
em is
help
ful i
n th
e pr
oces
s of
pro
blem
sol
ving
.
Th
e ro
ads
con
nec
tin
g th
e to
wn
s of
Kin
gs,
Ch
ana,
and
Hol
com
b fo
rm a
tri
angl
e.D
avis
Ju
nct
ion
is
loca
ted
in
th
e in
teri
or o
f th
is t
rian
gle.
Th
e d
ista
nce
s fr
omD
avis
Ju
nct
ion
to
Kin
gs,C
han
a,an
d H
olco
mb
are
3 k
m,
4 k
m,a
nd
5 k
m,r
esp
ecti
vely
.Jan
e be
gin
s at
Hol
com
b an
dd
rive
s d
irec
tly
to C
han
a,th
en t
o K
ings
,an
d t
hen
bac
k t
oH
olco
mb.
At
the
end
of
her
tri
p,s
he
figu
res
she
has
tra
vele
d25
km
alt
oget
her
.Has
sh
e fi
gure
d t
he
dis
tan
ce c
orre
ctly
?
To s
olve
thi
s pr
oble
m,a
dia
gram
can
be
draw
n.B
ased
on
this
dia
gram
and
the
Tri
angl
e In
equa
lity
The
orem
,the
di
stan
ce f
rom
Hol
com
b to
Cha
na is
less
tha
n 9
km.S
imila
rly,
the
dist
ance
fro
m C
hana
to
Kin
gs is
less
tha
n 7
km,a
nd t
hedi
stan
ce f
rom
Kin
gs t
o H
olco
mb
is le
ss t
han
8 km
.
The
refo
re,J
ane
mus
t ha
ve t
rave
led
less
tha
n (9
"7
"8)
km
or
24
km v
ersu
s he
r ca
lcul
ated
dis
tanc
e of
25
km.
Exp
lain
wh
y ea
ch o
f th
e fo
llow
ing
stat
emen
ts i
s tr
ue.
Dra
w a
nd
lab
el a
dia
gram
to
be u
sed
in
th
e ex
pla
nat
ion
.
1.If
an
alti
tude
is d
raw
n to
one
sid
e of
a t
rian
gle,
then
the
leng
th o
f th
e al
titu
de is
less
tha
n on
e-ha
lf t
he s
um o
f th
ele
ngth
s of
the
oth
er t
wo
side
s.
If B!
D!is
the
alti
tude
,the
n it
is t
rue
that
B!D!
$A!
C!.
Then
"B
DC
and
"B
DA
are
righ
t tr
iang
les.
By
Theo
rem
6-8
,BD
'B
C a
nd B
D'
BA
.Usi
ngTh
eore
m 6
-2,2
BD
'B
A!
BC
.Thu
s,B
D'
&1 2&(B
A!
BC
).
2.If
poi
nt Q
is in
the
inte
rior
of *
AB
Can
d on
the
ang
le b
isec
tor
of !
B,t
hen
Qis
equ
idis
tant
fro
m A!
B!an
d C!
B!.(
Hin
t:D
raw
Q!D!
and
Q!E!
such
tha
t Q!
D!$
A!B!
and
Q!E!
$C!
B!.)
If Q
is o
n th
e bi
sect
or o
f !
B,Q!
D!$
A!B!
,and
Q!
E!$
C!B!
,the
n "
QE
B$
"Q
DB
by H
A.T
hus,
Q!E!
$Q!
D!by
CP
CTC
,whi
ch m
eans
tha
t Q
iseq
uidi
stan
t fr
om A!
B!an
d C!
B!.
CE
B
A
Q
D
AD
C
B
King
s
Davi
sJu
nctio
n
Chan
aHo
lcom
b
3 km
5 km
4 km
En
rich
men
t
NA
ME
____
____
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____
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____
____
____
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____
__D
ATE
____
____
____
PE
RIO
D__
___
5-5
5-5
Exam
ple
Exam
ple
Answers (Lesson 5-5)
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