theorems 4 – 18 & more definitions, too!. page 104, chapter summary: concepts and procedures...
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Page 104, Chapter Summary: Concepts and Procedures
After studying this CHAPTER, you should be able to . . .
2.1 Recognize the need for clarity and concision in proofs2.1 Understand the concept of perpendicularity
2.2 Recognize complementary and supplementary angles
2.3 Follow a five-step procedure to draw logical conclusions
2.4 Prove angles congruent by means of four new theorems
2.5 Apply the addition properties of segments and angles2.5 Apply the subtraction properties of segments and angles
2.6 Apply the multiplication and division properties of segments and angles2.7 Apply the transitive properties of angles and segments
2.7 Apply the Substitution Property
2.8 Recognize opposite rays2.8 Recognize vertical angles
2
Chapter 2, Section 1: “Perpendicularity”
COORDINATES
ORIGIN
PERPENDICULAR
X-axis
Y-axis
After studying this SECTION, you should be able to . . .
Related Vocabulary
OBLIQUE LINES
3
2.1 Recognize the need for clarity and concision in proofs
2.1 Understand the concept of perpendicularity
After studying this SECTION, you should be able to . . .
Related Vocabulary
Chapter 2, Section 1: “Perpendicularity”
4
2.1 Recognize the need for clarity and concision in proofs
2.1 Understand the concept of perpendicularity
DEFINITIONS
PERPENDICULAR – lines, rays, or segments that INTERSECT at right angles
OBLIQUE LINES – when lines, rays, or segments INTERSECT and are NOT PERPENDICULAR
After studying this SECTION, you should be able to . . .
Related Vocabulary
Chapter 2, Section 1: “Perpendicularity”
5
2.1 Recognize the need for clarity and concision in proofs
2.1 Understand the concept of perpendicularity
PERPENDICULAR
If a right angle is created at the intersection of two rays, then the rays are perpendicular!CONDITIONAL
If two rays are perpendicular, then they create a right angle!CONVERSE
RIGHT ANGLENOT PERPENDICULAR
⊬H
KO
Given: OH OK
If OH OK ,
then ∡HOK is a Rt ∡and if ∡HOK is a Rt ∡,then m∡HOK = 90
SYMBOLS:
CHAIN REASONING
After studying this SECTION, you should be able to . . .
Related Vocabulary
Chapter 2, Section 1: “Perpendicularity”
6
2.1 Recognize the need for clarity and concision in proofs
2.1 Understand the concept of perpendicularity
If a right angle is created at the intersection of two rays, then the rays are perpendicular!CONDITIONAL
If two rays are perpendicular, then they create a right angle!CONVERSE
H
KO
Given: m∡ HOK = 90
then OH OK
then ∡HOK is a Rt ∡and if ∡HOK is a Rt ∡,If m∡HOK = 90
CHAIN REASONING
90⁰
Right Angle
s
90⁰
After studying this SECTION, you should be able to . . .
Chapter 2, Section 1: “Perpendicularity”
7
2.1 Recognize the need for clarity and concision in proofs
2.1 Understand the concept of perpendicularity
If a right angle is created at the intersection of two rays, then the rays are perpendicular!CONDITIONAL
If two rays are perpendicular, then they create a right angle!CONVERSE
H
KO
90⁰
90⁰
90⁰
Right ∡
Right ∡Perpendicularity, right angles, and
90⁰ measurements all go together!
After studying this SECTION, you should be able to . . .
Related Vocabulary
Chapter 2, Section 1: “Perpendicularity”
8
2.1 Recognize the need for clarity and concision in proofs2.1 Understand the concept of perpendicularity
COORDINATES
ORIGIN
x-axis 1 2 3 4 5
1
2
3
4
5
-1
-2
-3
-4
-5
-1-2-3-4-5 0
y-axis
A (3, 2)B (-3, 2)C (-3, -2)
E (0, 0)F (4, 0)
G (-4, 0)
H (0, 3)
J (0, -3)
D (3, -2)
COORDINATES
COORDINATES
COORDINATES
Remember: The x-axis is to the y-axis
H
FG
J
Can you name the lines?
Can you name the ‖ lines?‖ parallel
Couldany lines drawn be
“oblique lines”?
Find the area of rectangle PACE
9
1 2 3 4 5
1
2
3
4
5
-1
-2
-3
-4
-5
-1-2-3-4-5 0
Given: AP ‖ to the y-axis CE ‖ to the y-axis
C
P
A
E
2.1 Example
4
7
AreaRECT = (length)(width)
Width = |y – y|
Width = |2 – (-2)|
Width = |2 + 2|
Width = |4|
Length = |x – x|
Length = |3 – (-4)|
Length = |3 + 4|
Length = |7|
AreaRECT = (7 units)(width)
AreaRECT = 28 units2
(4 units)
Remember an important property of rectangles is
that BOTH pairs of opposite sides are congruent, and:
If two segments are congruent, then they
have the SAME measure!
After studying this SECTION, you should be able to . . .
COMPLEMENTCOMPLEMENTARY ANGLES
SUPPLEMENT
SUPPLEMENTARY ANGLES
Related Vocabulary
Chapter 2, Section 2: “Complementary and Supplementary Angles”
10
2.2 Recognize complementary and supplementary angles
(NOT the same as: “You look very nice today!”)
(NOT THE SAME AS: “Did you take your vitamins today!”)
After studying this SECTION, you should be able to . . .
Related Vocabulary
Chapter 2, Section 2: “Complementary and Supplementary Angles”
11
2.2 Recognize complementary and supplementary angles
COMPLEMENTCOMPLEMENTARY ANGLES
- the NAME given to each of the two angles whose sum equals 90⁰
- two angles whose sum equals a 90⁰ right angle
15⁰
75⁰
30⁰
60⁰57
41’20
”
⁰
3218
’40”
⁰
V
N V
NA A
V
N
QUESTION!
If two angles are COMPLEMENTARY
ANGLES,(then) are they also ADJACENT
ANGLES?
After studying this SECTION, you should be able to . . .
Related Vocabulary
Chapter 2, Section 2: “Complementary and Supplementary Angles”
12
2.2 Recognize complementary and supplementary angles
SUPPLEMENTSUPPLEMENTARY ANGLES
- the NAME given to each of the two angles whose sum equals 180⁰
- two angles whose sum equals a 180⁰ straight angle
85⁰
95⁰
130⁰
50⁰67 41’20”⁰112 18’40”⁰
T
R
T AR
A
T
R
P
QUESTION!
If two angles are SUPPLEMENTARY
ANGLES,(then) are they also ADJACENT
ANGLES?
After studying this SECTION, you should be able to . . .
Related Vocabulary
Chapter 2, Section 2: “Complementary and Supplementary Angles”
x
2x + 15
x + 2x + 15 = 90
13
2.2 Recognize complementary and supplementary angles
The measure of one of two complementary angles is 15 more than twice the other. Find the measure of each angle.
Write equation
THINK –
If two angles are complementary
angles,then their sum
equals 90! Simplify 3x + 15 =
90 Solve for x 3x = 75
x = 25 Substitute
25⁰
50 + 1575⁰
Is the answer reasonable?Is one of the
angles 15 more than twice the
other?
YES!
14
If a problem contains ONLY complements or ONLY supplements, use the previous method.
Begin by drawing a right angle for two complementary angles or a straight angle to model two supplementary angles,
and label them according to the information given in the problem!
HOWEVER, if a problem refers to BOTH the complement AND the supplement
in the same problem ,
use the NEXT method:
15
Chapter 2, Section 2: “Complementary and Supplementary Angles”After studying this SECTION, you should be able to . . .
2.2 Recognize complementary and supplementary angles
Use the “Boxer” Method to write expressions for each type of angle:
Are you wondering, “what is the “Boxer Method”?”
Well, first make a “BOX,” and then let “the angle” equal x
THE ANGLE
COMPLEMENT
SUPPLEMENT
x⁰
(90 – x)⁰
(180 – x)⁰
x⁰
x⁰30⁰
30⁰
30⁰
60⁰
150⁰
60⁰
150⁰
Complements
Supplements
16
Chapter 2, Section 2: “Complementary and Supplementary Angles”Example 2.2 Recognize complementary and supplementary
anglesThe measure of the supplement of an angle is 60 less than 3 times the
complement of the angle.
Find the measure of the complement.The measure of the supplement of an angle is 60 less than 3 times the complement
ANGLE
COMP
SUPP
x
90 – x
180 – x Complement
Supplement
“the angle”x
90 – x
180 – x
(180 – x) = 3(90 – x) - 60
180 – x = 270 -3x -60 180 + 2x = 210
2x = 30
x = 15
15⁰
75⁰
165⁰
√
15
15
15180 – 15
90 – 15
x
x
75⁰
Related Vocabulary
After studying this SECTION, you should be able to . . .
Chapter 2, Section 3: “Drawing Conclusions”
No NEW vocabulary!
17
2.3 Follow a five-step procedure to draw logical conclusions
See very important TABLE on page 72!
After studying this SECTION, you should be able to . . .
Chapter 2, Section 3: “Drawing Conclusions”
NOTE: The “If . . .” part of the reason should match the GIVEN information!
5-STEP Procedure for Drawing Conclusions:
• 1. MEMORIZE theorems, definitions, and postulates
• 2. Look for KEY WORDS and SYMBOLS in the “givens”
• 3. Think of all the theorems, definitions, and postulates that involve those keys.
• 4. Decide which theorem, definition, or postulate allows you to draw a conclusion
• 5. DRAW A CONCLUSION, and give a reason to justify it.
18
2.3 Follow a five-step procedure to draw logical conclusions
AND the “then . . .” part matches the CONCLUSION being justified!CAUTION! Be sure not to reverse that order!!!
After studying this SECTION, you should be able to . . .
Chapter 2, Section 3: “Drawing Conclusions”
1) If B bisects AC, then ____?______
3) If ∡ABC ≅ ∡CBD ≅ ∡DBE, then ____?____.
2) If AB AC, then _____?_______.
D
E
BA C
19
PRACTICE EXAMPLES
B
B
A
AC
C
then . . . AB ≅ BC
then ∡BAC is a Rt ∡
then . . . BC and BD trisect ∡ABE
Key info: a point, bisect, and seg
Key info: ,, and
Key info: ∡ ≅ ∡ ≅ ∡
After studying this SECTION, you should be able to . . .
Chapter 2, Section 3: “Drawing Conclusions”
1) If B bisects AC, then ____?______
3) If ∡ABC ≅ ∡CBD ≅ ∡DBE,
then ____?____.
2) If AB AC, then _____?_______.
BA C
20
JUSTIFY your CONCLUSIONS!
B
A C
then . . . AB ≅ BC
then ∡BAC is a Rt ∡
then . . . BC and BD trisect ∡ABE
D
EB
AC
REASON: If a seg is bisected by a point, then the seg is divided into two congruent segs
REASON: If two rays are perpendicular, then they form a right angle
REASON: If an angle has been divided into 3 congruent angles,
then it was trisected by two rays.
Related Vocabulary
After studying this SECTION, you should be able to . . .
THEOREM #5
THEOREM #4
THEOREM #6
THEOREM #7
21
2.4 Prove angles congruent by means of four new theorems
Chapter 2, Section 4: “Congruent Supplements and Complements”
No NEW vocabulary!BUT . . .
THEOREM #4
After studying this SECTION, you should be able to . . .
If angles are supplementary to the same angle,
then they are congruent
22
Chapter 2, Section 4: “Congruent Supplements and Complements”
2.4 Prove angles congruent by means of four new theorems
120⁰ 1G
2
∡1 is supplementary to ∡G
∡2 is also supplementary to ∡G What can we conclude about ∡1 and ∡2?
60⁰ 60⁰
=
After studying this SECTION, you should be able to . . .
If angles are supplementary to congruent angles,
then they are congruent
23
Chapter 2, Section 4: “Congruent Supplements and Complements”
2.4 Prove angles congruent by means of four new theorems
G
∡G is supplementary to ∡E
∡O is supplementary to ∡M
What can we conclude about ∡G and ∡M?
∡E ≅ ∡OM
EO
50⁰ 50⁰130⁰
130⁰
THEOREM #5
After studying this SECTION, you should be able to . . .
If angles are complementary to congruent angles,
then they are congruent
24
Chapter 2, Section 4: “Congruent Supplements and Complements”
2.4 Prove angles congruent by means of four new theorems
If angles are complementary to the same angle,
then they are congruentWhat can we conclude?
What can we conclude? THEOREM #7
THEOREM #6
The only difference is the sum! (90 versus 180)
Complete a Proof!
After studying this SECTION, you should be able to . . .
Given:
PROVE:
Chapter 2, Section 4: “Congruent Supplements and Complements”
∡1 is comp to ∡4
25
R
S
V
∡2 is comp to ∡3RT bisects ∡SRVTR bisects ∡STV
1
2
3
4T
?
?
4) ∡3 ≅ ∡41) ∡1 is comp to ∡42) ∡2 is comp to ∡33) RT bisects ∡SRV
1) Given2) Given3) Given4) If a ray bis an ∡, it div it into 2 ≅ ∡s 5) ∡1 ≅ ∡2 5) If ∡’s comp ≅ ∡s, then they are ≅
Statements Reasons
6) TR bisects ∡STV 6) If an ∡ is div into 2 ≅ ∡s, then it was bisected by a ray!
2.5 Apply the addition properties of segments and angles
Related Vocabulary
After studying this SECTION, you should be able to . . .
Chapter 2, Section 5: “Addition and Subtraction Properties”
26
2.5 Apply the subtraction properties of segments and angles
7cm3cm7 cmA B C D
AC = BD, because
AC BDAB + BC = BC + CD,
If two segments have the same measure, they are congruent!
(7) + (7)(3) = (3) +(Commutative Property of Addition!)
If a segment is added to two congruent segments, the sums are
congruent. (Addition Property)
Note that we first need to know that two segments are congruent, and then that we
are adding the SAME segment to both of them.
2.5 Apply the addition properties of segments and angles
Related Vocabulary
After studying this SECTION, you should be able to . . .
Chapter 2, Section 5: “Addition and Subtraction Properties”
27
2.5 Apply the subtraction properties of segments and angles
If two angles have the same measure, they are congruent!
(Commutative Property of Addition!)
mDBE = 50.03
mABC = 50.03A
B
E
C
D
ABD CBEm
∡ABC
= 5
0⁰
m∡DBE = 50⁰
50 + ∡CBD = ∡CBD + 50
m∡ABC + m∡CBD = m∡CBD + m∡DBE
m∡ABD = m∡CBE, so
If an angle is added to two congruent angles,
then the sums are congruent. (Addition Property)
Note that we first need to know that two angles are congruent, and then that we are
adding the SAME angle to both of them.
ÐABC @ ÐDBE
2.5 Apply the addition properties of segments and angles
Related Vocabulary
After studying this SECTION, you should be able to . . .
Chapter 2, Section 5: “Addition and Subtraction Properties”
28
2.5 Apply the subtraction properties of segments and angles
If two angles have the same measure, they are congruent!
mDBE = 50.03
mABC = 50.03A
B
E
C
D
m∡ABD = 80⁰
m∡CBE = 80⁰
80 - ∡CBD = ∡CBD
80 -
m∡ABD - m∡CBD = m∡CBE - m∡CBD
m∡ABC = m∡DBE, so
If an angle is subtracted from two congruent angles, the differences
are congruent. (Subtraction Property)
Note that we first need to know that two angles are congruent, and then that we are
subtracting the SAME angle from both of them.
D E
GC
H
F
Chapter 2, Section 5: “Addition and Subtraction Properties”
After studying this SECTION, you should be able to . . .
2.5 Apply the addition properties of segments and angles2.5 Apply the subtraction properties of segments and angles
CF + FG = DE + EH
CG = DH, so
CG DH≅
If congruent segments are added to congruent segments, the sums are
congruent. (Addition Property)
Note that first we need 2 congruent segments, then we need 2 different
congruent segments to ADD.
30
J
I K
L
JIK JKI
Chapter 2, Section 5: “Addition and Subtraction Properties”
After studying this SECTION, you should be able to . . .
2.5 Apply the addition properties of segments and angles2.5 Apply the subtraction properties of segments and angles
m∡JIL + m∡LIK = m∡LKI + m∡JKLIf congruent angles are added to
congruent angles, the sums are congruent.
(Addition Property)
Note that first we need 2 congruent angles, then we need to add two
different congruent angles
10
10
Q RB AQR - BR = BA - BR
Chapter 2, Section 5: “Addition and Subtraction Properties”
After studying this SECTION, you should be able to . . .
2.5 Apply the addition properties of segments and angles2.5 Apply the subtraction properties of segments and angles
QB RA≅
If a segment (or angle) is subtracted from congruent segments (or angles),
the differences are congruent. (Subtraction Property)
Note that we need to start with congruent angles or segments and
then subtract the same angle or segment from both.
If a segment (or angle) is subtracted from congruent segments (or angles), the
differences are congruent. (Subtraction Property)
Note that we need to start with congruent angles or segments and then subtract the same angle or
segment from both. mABD - mCBD = mCBE - mCBDmABD = 78
mCBE = 78D
C
E
B
A
ABC DBE
Chapter 2, Section 5: “Addition and Subtraction Properties”
After studying this SECTION, you should be able to . . .
2.5 Apply the addition properties of segments and angles2.5 Apply the subtraction properties of segments and angles
mSTV - mWTV = mUVT - mWVT
W mSTV = mUVT = 130mWTV =mWVT = 30T V
US
2.5 Apply the subtraction properties of segments and angles2.5 Apply the addition properties of segments and angles
After studying this SECTION, you should be able to . . .
Chapter 2, Section 5: “Addition and Subtraction Properties”
∡STW UVW≅ ∡
If congruent segments (or angles) are subtracted from congruent segments (or angles), the differences are congruent.
(Subtraction Property)Note that we start with congruent segments or
angles, and then subtract congruent segments or angles.
Using the Addition and Subtraction Properties
An addition property is used when the segments or angles in the conclusion are greater than those in the given information
A subtraction property is used when the segments or angles in the conclusion are smaller than those in the given information.
Theorem: If a segment is added to two congruent segments, the sums are congruent. (Addition Property)
Given:
Conclusion: PR QS
RQP S
5. If two segments have the same measure then they are congruent
5.
4. Addition of Segments4. PR = QS
3. Additive Property of Equality3. PQ + QR = RS + QR
2. If two segments are congruent, then they have the same measure
2. PQ = RS
1. Given1.
ReasonsStatements
PQ RS
PQ @ RS
PR @ QS
Given:
Conclusion:
GJ HK
GH JK G K
M
H J
Statements Reasons
1.
2.
GJ HK
GH JK
How to use this theorem in a proof:
2. If a segment is subtracted from congruent segments, then the resulting segments are congruent. (Subtraction)
1. Given
? ?
Multiplication Property
If segments (or angles) are congruent, then their like multiples are congruent.
Example: If B, C, F, and G are trisection points and
then by the Multiplication
Property.
A B C D E F G H
,EFAB
EHAD
Division Property
If segments (or angles) are congruent, then their like divisions are congruent.
C
SA
T
D
ZO
G
If ∡CAT ≅ ∡DOG, andthen, ∡CAS ≅ ∡DOZ by the division property
AS and OZ are angle bisectors
Using the Multiplication and Division Properties in Proofs
Look for the DOUBLE USE of the words midpoint, trisects, or bisects in the “Givens.”
Use MULTIPLICATION if what is Given is less than the Conclusion
Use DIVISION if what is Given is greater than the Conclusion
Example
Given: O is the midpoint of
R is the midpoint of
Prove: Statements Reasons
NSMPMPNS
NRMO
M O P
N R S
1. MP NS≅2. O is mdpt of MP3. MO OP≅4. R is mdpt of NS5. NR RS≅5. MO NR≅
1. Given2. Given3. A mdpt divides a seg into 2 segs≅4. Given4. Same as #35. If segs are , then their like divisions are ≅ ≅
(DIVISION PROPERTY)
2.7 Apply the Transitive Property of angles and segments
2.7 Apply the Substitution Property
Related Vocabulary
After studying this SECTION, you should be able to . . .
SUBSTITUTE
SUBSTITUTION
Chapter 2, Section 7: “Transitive and Substitution Properties”
41
Theorems
Theorem 16
Theorem 17
After studying this SECTION, you should be able to . . .
THEOREM:
CONCLUSION?
AB ≅ BC
42
Chapter 2, Section 7: “Transitive and Substitution Properties”
2.7 Apply the Transitive Property of angles and segments
2.7 Apply the Substitution Property
BC ≅ CD
BA
C D
AB ≅ CD
If segments are congruent to the SAME segment,
then they are congruent to each other.
After studying this SECTION, you should be able to . . .
THEOREM:
CONCLUSION?
If angles are congruent to the SAME angle,
then they are congruent to each other.
∡1 ≅ ∡2
43
Chapter 2, Section 7: “Transitive and Substitution Properties”
2.7 Apply the Transitive Property of angles and segments
2.7 Apply the Substitution Property
∡2 ≅ ∡3 2
1
3
∡1 ≅ ∡3
After studying this SECTION, you should be able to . . .
THEOREM:
CONCLUSION?
AB ≅ NM
44
Chapter 2, Section 7: “Transitive and Substitution Properties”
2.7 Apply the Transitive Property of angles and segments
2.7 Apply the Substitution Property
QR ≅ MPBA Q
PAB ≅ CD
If segments are congruent to congruent segments,
then they are congruent to each other.
N M
R
NM ≅ MP
After studying this SECTION, you should be able to . . .
THEOREM:
CONCLUSION?
If angles are congruent to congruent angles,
then they are congruent to each other.
∡7 ≅ ∡5
45
Chapter 2, Section 7: “Transitive and Substitution Properties”
2.7 Apply the Transitive Property of angles and segments
2.7 Apply the Substitution Property
∡6 ≅ ∡8 57
6
∡7 ≅ ∡8∡5 ≅ ∡68
46
Chapter 2, Section 7: “Transitive and Substitution Properties”After studying this SECTION, you should be able to . . .
2.7 Apply the Transitive Property of angles and segments
2.7 Apply the Substitution Property
12 3
Given:∡1 comps ∡2∡2 ≅ ∡3 m∡1 + m∡2 = 90m∡2 ≅ m∡3∴ m∡1 + m∡3 = 90By Substitution Property!
2.8 Recognize opposite rays
2.8 Recognize Vertical Angles
Related Vocabulary
After studying this SECTION, you should be able to . . .
Vertical Angles-
THEOREM 18
Chapter 2, Section 8: “Vertical Angles”
Opposite Rays - (definition) – collinear rays that share a common endpoint
(definition) – two angles whose sides are formed by opposite rays.
Vertical angles are CONGURENT!
47
and extend in opposite directions
2.8 Recognize opposite rays
2.8 Recognize Vertical Angles
Related Vocabulary
After studying this SECTION, you should be able to . . .
Chapter 2, Section 8: “Vertical Angles”
Opposite Rays - (definition) – collinear rays that share a common endpoint
48
and extend in opposite directions
Name the opposite rays:
1)A B C
2) 3)
F
HD
GE
KI
LH
J
BA
and
BC
EH
and
EG
ED
and
EF
IL
and
IJ
2.8 Recognize opposite rays
2.8 Recognize Vertical Angles
Related Vocabulary
After studying this SECTION, you should be able to . . .
Chapter 2, Section 8: “Vertical Angles”
49
4) Which numbered angle is vertical with ∡1?5) Which numbered angle is vertical with ∡4?6) If m∡1 = 65, find the measure of the numbered angles. F
HD
G
Vertical Angles-
(definition) – two angles whose sides are formed by opposite rays.
1 2
34
∡3∡2 65
°65°
115°115°
2.8 Recognize opposite rays
2.8 Recognize Vertical Angles
Related Vocabulary
After studying this SECTION, you should be able to . . .
Chapter 2, Section 8: “Vertical Angles”
50
7) If m∡3 = 55, which other numbered angle must be 55°?
Vertical Angles-
(definition) – two angles whose sides are formed by opposite rays.
∡6∡4 55
°
7) If m∡1 = 40, which other numbered angle must be 40°?
1
65
4
3240
°
51
A
F
E
C G H B
D
Conclusion: CE ≅ DB
E and D are the midpoints of AC and AB,
and AC ≅ AB
1. Given:
Because? AdditionSubtractionMultiplicationDivisionTransitiveSubstitution
Like divisions of ≅ segs are ≅
Self-Check Properties Quiz Questions
52
A
F
E
C G H B
D
Conclusion: CD ≅ EB
FE FD, and≅
FC ≅ FB
2. Given:
Because? AdditionSubtractionMultiplicationDivisionTransitiveSubstitution
≅SEGS added to ≅SEGS are ≅ SEGS
Self-Check Properties Quiz Questions
53
A
F
E
C G H B
D
Conclusion: ACD ABE∡ ≅ ∡
CD bisects ACB,∡
BE bisects ABC, ∡
3. Given:
Because?AdditionSubtractionMultiplicationDivisionTransitiveSubstitution
Like divisions of ≅ ∡s are ≅
and ACB ABC ∡ ≅
Self-Check Properties Quiz Questions
54
A
F
E
C G H B
D
Conclusion: CG BH≅
CG GH,≅
BH GH, ≅
4. Given:
Because?AdditionSubtractionMultiplicationDivisionTransitiveSubstitution
If segs are ≅ to SAME seg, then ≅ to each other
Self-Check Properties Quiz Questions
55
A
F
E
C G H B
D
Conclusion: ACD ABE∡ ≅ ∡
∡BCD CBE ,≅ ∡
5. Given:
Because?AdditionSubtractionMultiplicationDivisionTransitiveSubstitution
If ≅ ∡s are subtracted from ≅ ∡s,then the like diffs are ≅
and ACB ABC ∡ ≅
Self-Check Properties Quiz Questions
56
A
F
E
C G H B
D
Conclusion: FD + FB = EB
EF = FD, and
EF + FB = EB
6. Given:
Because? AdditionSubtractionMultiplicationDivisionTransitiveSubstitution
One seg measure can be substituted for the other in the EQUATION!
= !
Self-Check Properties Quiz Questions
57
A
F
E
C G H B
D
Conclusion: CG ≅ BH
CH BG≅
7. Given:
Because? AdditionSubtractionMultiplicationDivisionTransitiveSubstitution
If the SAME seg is subtracted from ≅SEGS , the like diffs are ≅
Self-Check Properties Quiz Questions
58
A
F
E
C G H B
D
Conclusion: 2( ABC) + CAB = ∡ ∡180°
∡CAB + ACB + ABC = ∡ ∡180°
8. Given:
Because?AdditionSubtractionMultiplicationDivisionTransitiveSubstitution
and ACB ABC ∡ ≅
One ANGLE measure can be substituted for the other in the EQUATION!
Self-Check Properties Quiz Questions
59
A
F
E
C G H B
D
Conclusion: ACB ABC∡ ≅ ∡
CD bisects ACB,∡
BE bisects ABC, ∡
9. Given:
Because?AdditionSubtractionMultiplicationDivisionTransitiveSubstitution
Like multiples of ≅ ∡s are ≅
and ACD ABE ∡ ≅
Self-Check Properties Quiz Questions