1-1 study guide and intervention - ladue school...

Study Guide and InterventionPoints, Lines, and Planes

NAME ______________________________________________ DATE ____________ PERIOD _____

1-11-1

© Glencoe/McGraw-Hill 1 Glencoe Geometry

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on

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Name Points, Lines, and Planes In geometry, a point is a location, a line containspoints, and a plane is a flat surface that contains points and lines. If points are on the sameline, they are collinear. If points on are the same plane, they are coplanar.

Use the figure to name each of the following.

a. a line containing point A

The line can be named as �. Also, any two of the three points on the line can be used to name it.

AB��, AC��, or BC��

b. a plane containing point D

The plane can be named as plane N or can be named using three noncollinear points in the plane, such as plane ABD, plane ACD, and so on.

Refer to the figure.

1. Name a line that contains point A.

2. What is another name for line m ?

3. Name a point not on AC��.

4. Name the intersection of AC�� and DB��.

5. Name a point not on line � or line m .

Draw and label a plane Q for each relationship.

6. AB�� is in plane Q.

7. ST�� intersects AB�� at P.

8. Point X is collinear with points A and P.

9. Point Y is not collinear with points T and P.

10. Line � contains points X and Y.

�Q

AB

S

P

TY

X

�

Pm

AB

EC

D

�

N

AB

C

DExampleExample

ExercisesExercises

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Homework 1.1


Points, Lines, and Planes in Space Space is a boundless, three-dimensional set ofall points. It contains lines and planes.

a. How many planes appear in the figure?

There are three planes: plane N , plane O, and plane P.

b. Are points A, B, and D coplanar?

Yes. They are contained in plane O.


1. Name a line that is not contained in plane N.

2. Name a plane that contains point B.

3. Name three collinear points.


4. How many planes are shown in the figure?

5. Are points B, E, G, and H coplanar? Explain.

6. Name a point coplanar with D, C, and E.

Draw and label a figure for each relationship.

7. Planes M andN intersect in HJ��.

8. Line r is in plane N , line s is in plane M, and lines r and sintersect at point J.

9. Line t contains point H and line t does not lie in plane M orplane N.

N

Ms

t

r

HJ

A B

CD

EF

G HI

J

N

A

C

B

D

E

NO

P

AB

CD

Study Guide and Intervention (continued)

Points, Lines, and Planes

NAME ______________________________________________ DATE ____________ PERIOD _____

1-11-1

ExampleExample

ExercisesExercises

John_2

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Homework 1.1

Skills PracticePoints, Lines, and Planes

NAME ______________________________________________ DATE ____________ PERIOD _____

1-11-1


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1. Name a line that contains point D.

2. Name a point contained in line n.

3. What is another name for line p?

4. Name the plane containing lines n and p.


5. Point K lies on RT��. 6. Plane J contains line s.

7. YP�� lies in plane B and contains 8. Lines q and f intersect at point Zpoint C, but does not contain point H. in plane U.



10. How many of the planes contain points F and E?

11. Name four points that are coplanar.

12. Are points A, B, and C coplanar? Explain.

WA

B

E

C

DF

U

q f

Z

HP

B

CY

J

sTR

K

G

n

A BD

C

p

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Homework 1.1



1. Name a line that contains points T and P.

2. Name a line that intersects the plane containing points Q, N, and P.

3. Name the plane that contains TN�� and QR��.


4. AK�� and CG�� intersect at point M 5. A line contains L(�4, �4) and M(2, 3). Line in plane T. q is in the same coordinate plane but does

not intersect LM��. Line q contains point N.



7. Name three collinear points.

8. Are points N, R, S, and W coplanar? Explain.

VISUALIZATION Name the geometric term(s) modeled by each object.

9. 10. 11.

12. a car antenna 13. a library card

strings

tip of pinSTOP

A M N

PW S

X R

QT

L

M

N

q

x

y

O

KT

M

C

GA

S

j

g

hT

M

N

Q

R

P

Practice Points, Lines, and Planes

NAME ______________________________________________ DATE ____________ PERIOD _____

1-11-1

John_2

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Homework 1.1

Study Guide and InterventionLinear Measure and Precision

NAME ______________________________________________ DATE ____________ PERIOD _____

1-21-2


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Measure Line Segments A part of a line between two endpoints is called a linesegment. The lengths of M�N� and R�S� are written as MN and RS. When you measure asegment, the precision of the measurement is half of the smallest unit on the ruler.

Find the length of M�N�.

The long marks are centimeters, and theshorter marks are millimeters. The length of M�N� is 3.4 centimeters. The measurement is accurate to within 0.5 millimeter, so M�N� isbetween 3.35 centimeters and 3.45centimeters long.

1cm 2 3 4

M N

Find the length of R�S�.

The long marks are inches and the short marks are quarter inches. The length of R�S�is about 1�

34� inches. The measurement is

accurate to within one half of a quarter inch,

or �18� inch, so R�S� is between 1�

58� inches and

1�78� inches long.

1 2in.

R S

Example 1Example 1 Example 2Example 2

ExercisesExercises

Find the length of each line segment or object.

1. 2.

3. 4.

Find the precision for each measurement.

5. 10 in. 6. 32 mm 7. 44 cm

8. 2 ft 9. 3.5 mm 10. 2�12� yd

1cm 2 31 2in.

1in.

S T

1cm 2 3

A B

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Homework 1.2


Calculate Measures On PQ��, to say that point M is between points P and Q means P, Q, and M are collinear and PM � MQ � PQ.

On AC��, AB � BC � 3 cm. We can say that the segments arecongruent, or A�B� � B�C�. Slashes on the figure indicate whichsegments are congruent.

AB

C

QMP


Linear Measure and Precision

NAME ______________________________________________ DATE ____________ PERIOD _____

1-21-2

Find EF.

Calculate EF by adding ED and DF.

ED � DF � EF1.2 � 1.9 � EF

3.1 � EF

Therefore, E�F� is 3.1 centimeters long.

E1.2 cm 1.9 cm

FD

Find x and AC.

B is between A and C.

AB � BC � ACx � 2x � 2x � 5

3x � 2x � 5x � 5

AC � 2x � 5 � 2(5) � 5 � 15

A x 2x CB

2x � 5


ExercisesExercises

Find the measurement of each segment. Assume that the art is not drawn to scale.

1. R�T� 2. B�C�

3. X�Z� 4. W�X�

Find x and RS if S is between R and T.

5. RS � 5x, ST � 3x, and RT � 48. 6. RS � 2x, ST � 5x � 4, and RT � 32.

7. RS � 6x, ST �12, and RT � 72. 8. RS � 4x, R�S� � S�T�, and RT � 24.

Use the figures to determine whether each pair of segments is congruent.

9. A�B� and C�D� 10. X�Y� and Y�Z�X

Y Z

3x � 5 5x � 1

9x2

A D

CB

11 cm

11 cm

5 cm5 cm

W Y

6 cm

XX ZY

3–4 in.31–

2 in.

A C

6 in.

23–4 in. BR T

2.0 cm 2.5 cm

S

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Homework 1.2

Study Guide and InterventionDistance and Midpoints

NAME ______________________________________________ DATE ____________ PERIOD _____

1-31-3


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Distance Between Two Points

Distance on a Number Line Distance in the Coordinate Plane

AB � |b � a | or |a � b |x

y

OC(1, –1)

A(–2, –1)

B(1, 3)

Pythagorean Theorem:

a2 � b2 � c2

Distance Formula:

d � �(x2 ��x1)2 �� (y2 �� y1)2�

A B

a b

Find AB.

AB � |(�4) � 2|� |� 6|� 6

�5 �4 �3 �2 �1 0 1 2 3

A B

Find the distance between A(�2, �1) and B(1, 3).


Pythagorean Theorem

(AB)2 � (AC)2 � (BC)2

(AB)2 � (3)2 � (4)2

(AB)2 � 25

AB � �25�� 5

Distance Formula

d � �(x2 ��x1)2 �� (y2 �� y1)2�AB � �(1 � (��2))2�� (3 �� (�1))�2�AB � �(3)2 �� (4)2�

� �25�� 5

Use the number line to find each measure.

1. BD 2. DG

3. AF 4. EF

5. BG 6. AG

7. BE 8. DE

Use the Pythagorean Theorem to find the distance between each pair of points.

9. A(0, 0), B(6, 8) 10. R(�2, 3), S(3, 15)

11. M(1, �2), N(9, 13) 12. E(�12, 2), F(�9, 6)

Use the Distance Formula to find the distance between each pair of points.

13. A(0, 0), B(15, 20) 14. O(�12, 0), P(�8, 3)

15. C(11, �12), D(6, 2) 16. E(�2, 10), F(�4, 3)

–10 –8 –6 –4 –2 0 2 4 6 8

A B C D E F G

ExercisesExercises

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Homework 1.2


Midpoint of a Segment

Midpoint on a If the coordinates of the endpoints of a segment are a and b,

Number Line then the coordinate of the midpoint of the segment is �a �2

b�.

Midpoint on a If a segment has endpoints with coordinates (x1, y1) and (x2, y2),

Coordinate Plane then the coordinates of the midpoint of the segment are ��x1 �

2x2�, �

y1 �

2y2��.

Find the coordinate of the midpoint of P�Q�.

The coordinates of P and Q are �3 and 1.

If M is the midpoint of P�Q�, then the coordinate of M is ��32� 1� � �

�22� or �1.

M is the midpoint of P�Q� for P(�2, 4) and Q(4, 1). Find thecoordinates of M.

M � ��x1 �

2x2�, �

y1 �

2y2�� 2

2� 4�, �

4 �2

1�� or (1, 2.5)

Use the number line to find the coordinate of the midpoint of each segment.

1. C�E� 2. D�G�

3. A�F� 4. E�G�

5. A�B� 6. B�G�

7. B�D� 8. D�E�

Find the coordinates of the midpoint of a segment having the given endpoints.

9. A(0, 0), B(12, 8) 10. R(�12, 8), S(6, 12)

11. M(11, �2), N(�9, 13) 12. E(�2, 6), F(�9, 3)

13. S(10, �22), T(9, 10) 14. M(�11, 2), N(�19, 6)

–10 –8 –6 –4 –2 0 2 4 6 8

A B C D E F G

–3 –2 –1 0 1 2

P Q


Distance and Midpoints

NAME ______________________________________________ DATE ____________ PERIOD _____

1-31-3

Example 1Example 1

Example 2Example 2

ExercisesExercises

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Homework 1.2

Reading to Learn MathematicsDistance and Midpoints

NAME ______________________________________________ DATE ____________ PERIOD _____

1-31-3


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1-3

Pre-Activity How can you find the distance between two points without a ruler?

Read the introduction to Lesson 1-3 at the top of page 21 in your textbook.

• Look at the triangle in the introduction to this lesson. What is the specialname for A�B� in this triangle?

• Find AB in this figure. Write your answer both as a radical and as adecimal number rounded to the nearest tenth.

Reading the Lesson

1. Match each formula or expression in the first column with one of the names in thesecond column.

a. d � �(x2 ��x1)2 �� ( y2 �� y1)2� i. Pythagorean Theorem

b. �a �

2b

� ii. Distance Formula in the Coordinate Plane

c. XY � |a � b| iii. Midpoint of a Segment in the Coordinate Plane

d. c2 � a2 � b2 iv. Distance Formula on a Number Line

e. ��x1 �

2x2

�, �y1 �

2y2

�� v. Midpoint of a Segment on a Number Line

2. Fill in the steps to calculate the distance between the points M(4, �3) and N(�2, 7).

Let (x1, y1) � (4, �3). Then (x2, y2) � ( , ).

d � ��( � )2 � ( � )2

MN � ��( � )2 � ( � )2

MN � ��( )2 � ( )2

MN � ��

MN � ��Find a decimal approximation for MN to the nearest hundredth.

Helping You Remember

3. A good way to remember a new formula in mathematics is to relate it to one you alreadyknow. If you forget the Distance Formula, how can you use the Pythagorean Theorem tofind the distance d between two points on a coordinate plane?

John_2

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Homework 1.2

Study Guide and InterventionAngle Measure

NAME ______________________________________________ DATE ____________ PERIOD _____

1-41-4


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Measure Angles If two noncollinear rays have a common endpoint, they form an angle. The rays are the sides of the angle.The common endpoint is the vertex. The angle at the right can be named as �A, �BAC, �CAB, or �1.

A right angle is an angle whose measure is 90. An acute anglehas measure less than 90. An obtuse angle has measure greater than 90 but less than 180.

A C

B

1

a. Name all angles that have R as avertex.Three angles are �1, �2, and �3. Forother angles, use three letters to namethem: �SRQ, �PRT, and �SRT.

b. Name the sides of �1.

RS��, RP��

S R T

PQ

1 2 3

Measure each angle andclassify it as right, acute, or obtuse.

a. �ABDUsing a protractor, m�ABD � 50.50 � 90, so �ABD is an acute angle.

b. �DBCUsing a protractor, m�DBC � 115.180 � 115 � 90, so �DBC is an obtuseangle.

c. �EBCUsing a protractor, m�EBC � 90.�EBC is a right angle.

BA

D E

C


ExercisesExercises


1. Name the vertex of �4.

2. Name the sides of �BDC.

3. Write another name for �DBC.

Measure each angle in the figure and classify it as right,acute, or obtuse.

4. �MPR

5. �RPN

6. �NPS

P

NM

R

S

C

BA

12

34

D

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Homework 1.3

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Congruent Angles Angles that have the same measure are congruent angles. A ray that divides an angle into two congruent angles is called an angle bisector. In the figure, PN�� is the angle bisector of �MPR. Point N lies in the interior of �MPR and �MPN � �NPR.

Refer to the figure above. If m�MPN � 2x � 14 and m�NPR � x � 34, find x and find m�MPR.Since PN�� bisects �MPR, �MPN � �NPR, or m�MPN � m�NPR.

2x � 14 � x � 34 m�NPR � (2x � 14) � (x � 34)2x � 14 � x � x � 34 � x � 54 � 54

x � 14 � 34 � 108x � 14 � 14 � 34 � 14

x � 20

QS�� bisects �PQT, and QP�� and QR�� are opposite rays.

1. If m�PQT � 60 and m�PQS � 4x � 14, find the value of x.

2. If m�PQS � 3x � 13 and m�SQT � 6x � 2, find m�PQT.

BA�� and BC�� are opposite rays, BF�� bisects �CBE, and BD�� bisects �ABE.

3. If m�EBF � 6x � 4 and m�CBF � 7x � 2, find m�EBC.

4. If m�1 � 4x � 10 and m�2 � 5x, find m�2.

5. If m�2 � 6y � 2 and m�1 � 8y � 14, find m�ABE.

6. Is �DBF a right angle? Explain.

BA C

F

ED

12 3

4

QP R

TS

P R

NM


Angle Measure

NAME ______________________________________________ DATE ____________ PERIOD _____

1-41-4

ExampleExample

ExercisesExercises

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Homework 1.3

Skills PracticeAngle Measure

NAME ______________________________________________ DATE ____________ PERIOD _____

1-41-4


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For Exercises 1–12, use the figure at the right.

Name the vertex of each angle.

1. �4 2. �1

3. �2 4. �5

Name the sides of each angle.

5. �4 6. �5

7. �STV 8. �1

Write another name for each angle.

9. �3 10. �4

11. �WTS 12. �2

Measure each angle and classify it as right, acute,or obtuse.

13. �NMP 14. �OMN

15. �QMN 16. �QMO

ALGEBRA In the figure, BA�� and BC�� are opposite rays,BD�� bisects �EBC, and BF�� bisects �ABE.

17. If m�EBD � 4x � 16 and m�DBC � 6x � 4,find m�EBD.

18. If m�ABF � 7x � 8 and m�EBF � 5x � 10,find m�EBF.

B CA

FE

D

M NL

Q O

P

U

T

VW

S5 3

21

4

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Homework 1.3

Reading to Learn MathematicsAngle Measure

NAME ______________________________________________ DATE ____________ PERIOD _____

1-41-4


Less

on

1-4

Pre-Activity How big is a degree?

Read the introduction to Lesson 1-4 at the top of page 29 in your textbook.• A semicircle is half a circle. How many degrees are there in a

semicircle?• How many degrees are there in a quarter circle?

Reading the Lesson1. Match each description in the first column with one of the terms in the second column.

Some terms in the second column may be used more than once or not at all.a. a figure made up of two noncollinear rays with a 1. vertex

common endpoint 2. angle bisectorb. angles whose degree measures are less than 90 3. opposite raysc. angles that have the same measure 4. angled. angles whose degree measures are between 90 and 180 5. obtuse anglese. a tool used to measure angles 6. congruent anglesf. the common endpoint of the rays that form an angle 7. right anglesg. a ray that divides an angle into two congruent angles 8. acute angles

9. compass10. protractor

2. Use the figure to name each of the following.a. a right angleb. an obtuse anglec. an acute angled. a point in the interior of �EBCe. a point in the exterior of �EBAf. the angle bisector of �EBCg. a point on �CBEh. the sides of �ABFi. a pair of opposite raysj. the common vertex of all angles shown in the figurek. a pair of congruent anglesl. the angle with the greatest measure

Helping You Remember3. A good way to remember related geometric ideas is to compare them and see how they

are alike and how they are different. Give some similarities and differences betweencongruent segments and congruent angles.

B G

28�28�

A

EF

CD

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Homework 1.3

Study Guide and InterventionAngle Relationships

NAME ______________________________________________ DATE ____________ PERIOD _____

1-51-5


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Pairs of Angles Adjacent angles are angles in the same plane that have a commonvertex and a common side, but no common interior points. Vertical angles are twononadjacent angles formed by two intersecting lines. A pair of adjacent angles whosenoncommon sides are opposite rays is called a linear pair.

Identify each pair of angles as adjacent angles, vertical angles,and/or as a linear pair.

ExampleExample

a.

�SRT and �TRU have a commonvertex and a common side, but nocommon interior points. They areadjacent angles.

c.

�6 and �5 are adjacent angles whosenoncommon sides are opposite rays.The angles form a linear pair.

D

CBA

5 6

RU

TS

b.

�1 and �3 are nonadjacent angles formedby two intersecting lines. They are verticalangles. �2 and �4 are also vertical angles.

d.

�A and �B are two angles whose measureshave a sum of 90. They are complementary.�F and �G are two angles whose measureshave a sum of 180. They are supplementary.

AB

FG

30�

60�

60�120�

N

RP

SM

14

32

ExercisesExercises

Identify each pair of angles as adjacent, vertical, and/or as a linear pair.

1. �1 and �2 2. �1 and �6

3. �1 and �5 4. �3 and �2

For Exercises 5–7, refer to the figure at the right.

5. Identify two obtuse vertical angles.

6. Identify two acute adjacent angles.

7. Identify an angle supplementary to �TNU.

8. Find the measures of two complementary angles if the difference in their measures is 18.

R S

NU

T

V

R

S

TU

V

P

Q5

432

1 6

John_2

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Homework 1.4


Perpendicular Lines Lines, rays, and segments that form four right angles are perpendicular. The right angle symbol indicates that the lines are perpendicular. In the figure at the right, AC�� is perpendicular to BD��,or AC�� ⊥ BD��.

Find x so that D�Z� ⊥ P�Z�.If D�Z� ⊥ P�Z�, then m�DZP � 90.

m�DZQ � m�QZP � m�DZP Sum of parts � whole

(9x � 5) � (3x � 1) � 90 Substitution

12x � 6 � 90 Simplify.12x � 84 Subtract 6 from each side.

x � 7 Divide each side by 12.

1. Find x and y so that NR�� ⊥ MQ��.

2. Find m�MSN.

3. m�EBF � 3x � 10, m�DBE � x, and BD�� ⊥ BF��. Find x.

4. If m�EBF � 7y � 3 and m�FBC � 3y � 3, find y so that EB�� ⊥ BC��.

5. Find x, m�PQS, and m�SQR.

6. Find y, m�RPT, and m�TPW.

P

S

V

R

W

T(4y � 5)�

(2y � 5)�

Q R

P S3x �

(8x � 2)�

B CA

DE

F

M

N

R

S Q

P

x �5x �

(9y � 18)�

Z

D

P

Q(9x � 5)�

(3x � 1)�

B

CD

A


Angle Relationships

NAME ______________________________________________ DATE ____________ PERIOD _____

1-51-5

ExampleExample

ExercisesExercises

John_2

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Homework 1.4

Skills PracticeAngle Relationships

NAME ______________________________________________ DATE ____________ PERIOD _____

1-51-5


Less

on

1-5

For Exercises 1–6, use the figure at the right and a protractor.

1. Name two acute vertical angles.

2. Name two obtuse vertical angles.

3. Name a linear pair.

4. Name two acute adjacent angles.

5. Name an angle complementary to �EKH.

6. Name an angle supplementary to �FKG.

7. Find the measures of an angle and its complement if one angle measures 18 degreesmore than the other.

8. The measure of the supplement of an angle is 36 less than the measure of the angle.Find the measures of the angles.

ALGEBRA For Exercises 9–10, use the figure at the right.

9. If m�RTS � 8x � 18, find x so that TR�� ⊥ TS��.

10. If m�PTQ � 3y � 10 and m�QTR � y, find y so that �PTR is a right angle.

Determine whether each statement can be assumed from the figure. Explain.

11. �WZU is a right angle.

12. �YZU and �UZV are supplementary.

13. �VZU is adjacent to �YZX.

ZX

W

V

Y

U

TP

QR

S

K

G

J

FE

H

John_2

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Homework 1.4

Reading to Learn MathematicsAngle Relationships

NAME ______________________________________________ DATE ____________ PERIOD _____

1-51-5


Less

on

1-5

Pre-Activity What kinds of angles are formed when streets intersect?

Read the introduction to Lesson 1-5 at the top of page 37 in your textbook.

• How many separate angles are formed if three lines intersect at a commonpoint? (Do not use an angle whose interior includes part of another angle.)

• How many separate angles are formed if n lines intersect at a commonpoint? (Do not count an angle whose interior includes part of another angle.)

Reading the Lesson1. Name each of the following in the figure at the right.

a. two pairs of congruent angles

b. a pair of acute vertical angles

c. a pair of obtuse vertical angles

d. four pairs of adjacent angles

e. two pairs of vertical angles

f. four linear pairs

g. four pairs of supplementary angles

2. Tell whether each statement is always, sometimes, or never true.

a. If two angles are adjacent angles, they form a linear pair.

b. If two angles form a linear pair, they are complementary.

c. If two angles are supplementary, they are congruent.

d. If two angles are complementary, they are adjacent.

e. When two perpendicular lines intersect, four congruent angles are formed.

f. Vertical angles are supplementary.

g. Vertical angles are complementary.

h. The two angles in a linear pair are both acute.

i. If two angles form a linear pair, one is acute and the other is obtuse.

3. Complete each sentence.

a. If two angles are supplementary and x is the measure of one of the angles, then themeasure of the other angle is .

b. If two angles are complementary and x is the measure of one of the angles, then themeasure of the other angle is .

Helping You Remember4. Look up the nonmathematical meaning of supplementary in your dictionary. How can

this definition help you to remember the meaning of supplementary angles?

65� 2 34

1

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Study Guide and InterventionPolygons

NAME ______________________________________________ DATE ____________ PERIOD _____

1-61-6


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Polygons A polygon is a closed figure formed by a finite number of coplanar linesegments. The sides that have a common endpoint must be noncollinear and each sideintersects exactly two other sides at their endpoints. A polygon is named according to itsnumber of sides. A regular polygon has congruent sides and congruent angles. A polygoncan be concave or convex.

Name each polygon by its number of sides. Then classify it asconcave or convex and regular or irregular.

ExampleExample

a.

The polygon has 4 sides, so it is a quadrilateral.It is concave because part of D�E� or E�F� lies in theinterior of the figure. Because it is concave, itcannot have all its angles congruent and so it isirregular.

c.

The polygon has 5 sides, so it is a pentagon. It isconvex. All sides are congruent and all angles arecongruent, so it is a regular pentagon.

D F

G

E b.

The figure is not closed, so it isnot a polygon.

d.

The figure has 8 congruent sidesand 8 congruent angles. It isconvex and is a regular octagon.

H

J K

LI

ExercisesExercises

Name each polygon by its number of sides. Then classify it as concave or convexand regular or irregular.

1. 2. 3.

4. 5. 6.

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Perimeter The perimeter of a polygon is the sum of the lengths of all the sides of thepolygon. There are special formulas for the perimeter of a square or a rectangle.

Write an expression or formula for the perimeter of each polygon.Find the perimeter.


Polygons

NAME ______________________________________________ DATE ____________ PERIOD _____

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ExampleExample

a.

P � a � b � c� 3 � 4 � 5� 12 in.

b.

P � 4s� 4(5)� 20 cm

c.

P � 2� � 2w� 2(3) � 2(2)� 10 ft

3 ft

2 ft

�

�

w w

5 cm

5 cm 5 cm

5 cm

s

s

s s3 in.5 in.

4 in.

a

b

c

ExercisesExercises

Find the perimeter of each figure.

1. 2.

3. 4.

Find the length of each side of the polygon for the given perimeter.

5. P � 96 6. P � 48x

x

2x

x � 2

rectangle

2x

x

1 cm

24 yd

19 yd

12 yd 14 yd

27 yd

square

5.5 ft

3.5 cm

3 cm2.5 cm

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Skills PracticePolygons

NAME ______________________________________________ DATE ____________ PERIOD _____

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Name each polygon by its number of sides and then classify it as convex orconcave and regular or irregular.

1. 2. 3.

4. 5. 6.


7. 8. 9.

COORDINATE GEOMETRY Find the perimeter of each polygon.

10. triangle ABC with vertices A(3, 5), B(3, 1), and C(0, 1)

11. quadrilateral QRST with vertices Q(�3, 2), R(1, 2), S(1, �4), and T(�3, �4)

12. quadrilateral LMNO with vertices L(�1, 4), M(3, 4), N(2, 1), and O(�2, 1)

ALGEBRA Find the length of each side of the polygon for the given perimeter.

13. P � 104 millimeters 14. P � 84 kilometers 15. P � 88 feet4w � 1

w

10 in.

10 in.2 in. 2 in.

2 in.2 in.2 in. 2 in.

5 m2 m 3 m

6 m4 m

40 yd

20 yd

18 yd 20 yd

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Name each polygon by its number of sides and then classify it as convex orconcave and regular or irregular.

1. 2. 3.


4. 5. 6.

COORDINATE GEOMETRY Find the perimeter of each polygon.

7. quadrilateral OPQR with vertices O(�3, 2), P(1, 5), Q(6, 4), and R(5, �2)

8. pentagon STUVW with vertices S(0, 0), T(3, �2), U(2, �5), V(�2, �5), and W(�3, �2)

ALGEBRA Find the length of each side of the polygon for the given perimeter.

9. P � 26 inches 10. P � 39 centimeters 11. P � 89 feet

SEWING For Exercises 12–13, use the following information.Jasmine plans to sew fringe around the scarf shown in the diagram.

12. How many inches of fringe does she need to purchase?

13. If Jasmine doubles the width of the scarf, how many inches of fringe will she need?

16 in.

4 in. 4 in.

16 in.

2x � 2

x � 9

5x � 4

3x � 5

2x � 3

6n � 8

n

14 cm

14 cm

4 cm

4 cm 6 cm6 cm

6 cm2 cm

32 mi

33 mi21 mi

7 mm

10 mm18 mm

18 mm

Practice Polygons

NAME ______________________________________________ DATE ____________ PERIOD _____

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Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

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Perimeter and Area of Irregular ShapesTwo formulas that are used frequently in mathematics are perimeter andarea of a rectangle.

Perimeter: P � 2� � 2wArea: A � �w, where � is the length and w is the width

However, many figures are combinations of two or more rectangles creatingirregular shapes. To find the area of an irregular shape, it helps to separatethe shape into rectangles, calculate the formula for each rectangle, then findthe sum of the areas.

Find the area of thefigure at the right.Separate the figure into two rectangles.

A � �wA1 � 9 2 A2 � 3 3

� 18 � 9

18 � 9 � 27

The area of the irregular shape is 27 m2.

Find the area and perimeter of each irregular shape.

1. 2.

3.4.

For Exercises 5–8, find the perimeter of the figures in Exercises 1–4.

5. 6. 7. 8.

9. Describe the steps you used to find the perimeter in Exercise 1.

9 ft

2 ft

3 ft

7 ft

6 ft

4 ft

8 cm

4 cm

2 cm

2 cm

4 cm

4 cm

6 cm

4 cm

9 m

26 m

6 m13 m

7 m

12 m

2 in.

4 in. 4 in.

1 in.

9 m

3 m

1

25 m

2 m

9 m

3 m

5 m

2 m

ExampleExample

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• •

Foundations of GeometrySection 1.1 The basic geometric objects: point, line, plane.

Object point line plane

Symbols A, B, C, D, ... AB , CD , l, k, m ABC , ABD , N , M

Picture

Equality A sB s C sD l = AB s CD ABC = ABD = N

Congruence A yB y C yD l y AB y CD ABC y ABD y N

Propertiestwo geometric objects are congruent just in case they are geometrically equivalent as objects, i.e. one can be obtained from the other by rigid motions using only translations and rotations, but not using stretches or shrinksevery geometric object is geometrically equivalent (congruent) to itselfall points are geometrically equivalent (congruent) but not necessarily equalall lines are geometrically equivalent (congruent) but not necessarily equalall planes are geometrically equivalent (congruent) but not necessarily equaltwo distinct points determine a unique linethree noncollinear points determine a unique plane

Examples: Intersections of Geometric Objects

Line l = AC intersects line

m = BD in point B.

Line AE intersects plane N in point B.

Planes P , O , and N all intersect

in line AB .

• •

• • • •

• • • • • •

• •

Section 1.2 More geometric objects: (line) segment, ray, angle.

Object segment ray angle

Symbols AB___

, DA___

, BC___

AB , BA , BC , DC :ADE,:AEC,:EBC

Picture

Equality AB___

s DA___

s BC___

AB s BA s BC :ADE s:AEC s:EBC

Congruence AB___

“ DA___

“ BC___

AB y BA y BC :ADE “:AEC “:EBC

Propertiessegments have two (finite) endpoints and are not in general congruent, since the have a measurerays have one (finite) endpoint and one infinite extentall rays are geometrically equivalent (congruent) but not necessarily equalan angle is the union of two noncollinear rays with the same endpointangles are not in general congruent, since they have a measuretwo distinct points determine a unique seqmenttwo distinct points with choice of endpoint determine a unique ray

Examples: Types of Angles

:ABD is obtuse, :DBC is acute, together they are supplementary and also a linear pair, B is the vertex of both angles

:GCH and :BCD are vertical angles, and so are :CBE and

:ABF, CH and CF are opposite rays

:EBF and :FBC are adjacent and complementary, :EBF and :EBD are adjacent

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• •

• •

• •

• •

Section 1.3 More geometric objects: triangles, quadrilaterals, polygons.

Object triangle quadrilateral polygon

Symbols 6ABC, 6NPM, 6TQU ABCD, PQRS, MNTU P1 P2 P3 ... Pn

Picture

Equality 6ABC =6ABC s 6NPM

PQRS = PQRS sMNTU P1 P2 ... Pn = P1 P2 ... Pn

Congruence 6ABC y6ABC “ 6NPM

PQRS yPQRS “MNTU P1 P2 ... Pn yP1 P2 ... Pn

Propertiesevery polygon has at least three sides and is a closed figure in the planea triangle (3-gon) is the simplest polygon and next simplest is a quadrilateral (4-gon)no three vertices of a polygon are collinear the segment connecting any two interior points of a convex polygon is completely contained within the interior of the polygona polygon is concave just in case it is not convex, i.e. it has a "dent"a regular polygon has all sides congruent and all angles congruent

Examples: Types of Polygons

a convex, regular polygon, a hexagon or 6-gon

a concave octagon, an 8-gon, notregular, and not convex

a concave polygon, an n-gon, notregular, and not convex

1-1 study guide and intervention - ladue school...

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