SECTION Ready to Go On? Skills Intervention 4A 4-1 ... angles have a sum of 90 . ... interior angle exterior angle remote interior angle ... Ready to Go On? Skills Intervention

Find these vocabulary words in Lesson 4-1 and the Multilingual Glossary.

Classifying Triangles by Angle Measures

A. �PQS

A right angle has a measure of .

Since �QPS is a angle, �PQS is a triangle.

B. �PRQ

First, find m�QRP. Since �QRP and �SRP form a pair, the angles are

. To find m�QRP, subtract 54� from .

m�QRP � � 54� �

What kind of angle is �QRP ? . So, �PRQ is an triangle.

C. �PRS

First, find m�RPS.

Since �RPS and �RPQ form a angle, the angles are

.

To find m�RPS, subtract 23� from .

m�RPS � � 23� �

What kind of angle is �RPS? .

What kind of angles are �SRP and �PSR ?

So, �PRS is an triangle.

Classifying Triangles by Side Lengths

A. �JKL

How many sides are congruent in �JKL?

What kind of triangle is �JKL?

B. �KML

Find KL. How many sides are congruent, or have the same measure, in

�KML? So, what kind of triangle is �KML?

4AReady to Go On? Skills Intervention4-1 Classifying Triangles

Vocabulary

acute triangle equiangular triangle right triangle obtuse triangle

equilateral triangle isosceles triangle scalene triangle

Q

P

R

S59°

54°23°

31°

K

JM

L7 4

98

Copyright © by Holt, Rinehart and Winston. 41 Holt GeometryAll rights reserved.

Name Date Class

SECTION

Complementary angles have a sum of 90�.

A roofer is making repairs on the roof of a house. In order to be safe, he sets his ladder so that it makes a 15� angle with the house. What angle will his ladder make with the ground?

Understand the Problem

1. What angle does the ladder make with the house?

2. What angle does the house form with the ground?

3. What kind of triangle is �ABC ?

Mark the figure with the information given in the problem.

Make a Plan

4. The acute angles of a right triangle are complementary, so the sum of the

measures of the acute angles equals .

5. Complete: m� � m�C � 90�

6. Write an equation by substituting the known angle measures. � m� � 90�

Solve

7. Solve the equation you wrote in Exercise 6:

� m�C � 90�

� � Subtract 15 from both sides to isolate the variable.

m�C �

8. What angle does the roofer’s ladder make with the ground?

Look BackYou can check your work in two ways.

9. What is the sum of the angles in a triangle?

10. From your answer in Exercise 9, you know that m�A � m� � m� � .

11. Substitute the angle measures and check your work. 90� � 15� � �

Does your answer check?

12. To check using a second method, substitute your solution from Exercise 8

into the equation you wrote in Exercise 6: 15� � � 90�

Does your answer check?

4AReady to Go On? Problem Intervention4-2 Angle Relationships in Triangles

B

C Aground

ladder house

Name Date Class


SECTION


Finding Angle Measures in TrianglesFind m�C.

By the Triangle Sum Theorem, the sum of the angle

measures in a triangle is .

In this triangle, m�A � m� � m� � 180�.

108� � � m�C � 180� Substitute known measures.

� m�C � 180� Add.

� � Subtract to isolate the variable.

m�C � Solve.

Finding Angle Measures in Right TrianglesOne of the acute angles in a right triangle measures 37.9�. What is the measure of the other acute angle?

Let the acute angles be �T and �U, with m�T � 37.9�.Since the measures of the acute angles in a right triangle are complementary,

m�T � m�U � . Substitute 37.9� for m�T and solve for m�U.

� m�U � �

� � Subtract to isolate the variable.

m�U � � Solve.

Applying the Exterior Angles TheoremFind m�Q.

Using the Exterior Angles Theorem, m� � m� � m�PRS. Substitute the given angle measures into the equation and solve for x.

� (5x � 3) �

� 5x � Add.

5x � 8x � Subtract 47 from both sides.

� �42 Subtract 8x from both sides.

x � Divide both sides by �3.

Substitute the value of x into (5x � 3) to find m�Q: (5x � 3) � (5)( ) � 3 �

4AReady To Go On? Skills Intervention4-2 Angle Relationships in Triangles

Vocabulary

Auxiliary line corollary interior exterior

interior angle exterior angle remote interior angle

108° 24°24°AA BB

CC

U

T

(8(8xx + 5)°+ 5)°xx (5(5xx + 3)°+ 3)°xx

44°44°

RRSS QQ

PP

4


Name Date Class

SECTION

Ready To Go On? Skills Intervention4-3 Congruent Triangles


Naming Congruent Corresponding PartsGiven �BCD � �PQR. Identify the congruent corresponding parts to �B and

_ BD .

In a congruence statement, vertices are written in corresponding .

�B corresponds with � , so �B � � .

_

BD corresponds with , so _

BD � .

Using Corresponding Parts of Congruent TrianglesGiven �DEF � �WXY.

A. Find the value of m.

�D corresponds with � , so �D � � .

Since �D � � , m�D � m� .

Substitute values for the angle measures D and W. Solve to find the value of m.

87 �

�2 �2 Subtract 2 from both sides.

85 �

85 ___ 5 � _____ 5 Divide both sides by 5.

� m Solve for m.

B. Find DE.

First find the value of x. _

XY corresponds with , so _

XY � and XY � . XY � EF Substitute values for XY and EF and solve for x.

3x � 7 �

�7 �7 Add 7 to both sides.

3x �

3x ___ 3 � _____ 3 Divide both sides by 3.

x � Solve for x.

Substitute the value of x into DE and simplify. DE � 2x � 9 � 2( )� 9 �

4A

Vocabulary

corresponding angles corresponding sides congruent polygons

DD

EE FF26

WW

YYXX

2x – 9 87°

(5m + 2)°

3x – 7

Name Date Class


SECTION


Name Date Class

4-1 Classifying TrianglesClassify each triangle by its angle measures.

1. �QPR

2. �SRQ

3. �TRQ

Classify each triangle by its side lengths.

4. �QNM

5. �MPQ

6. �NLM

4-2 Angle Relationships in TrianglesFind each angle measure.

7. m�GFC 8. m�BAC

(8x + 19)°(12x – 7)°

26°

G F

C

D

(15x + 14)°(9x – 28)°

A C

B

D

9. A high school baseball team is designing a pennant with the school logo. The pennant is an isosceles triangle and the measure of the vertex angle is 46�.

Find the measure of the base angles.

Ready to Go On? Quiz4A

SECTION

60° 60°

30°

30°

T

P SR

Q

8

8

= =8

L P Q N

M

46°

4-3 Congruent TrianglesGiven �MNO � �GHI. Identify the congruent corresponding parts.

10. _

MO � 11. _

GH �

12. �N � 13. �G �

Given �ABC � �LMN. Find each value.

14. LM

15. x

16. Given: ‹

__ › RS �

‹

__ › UT , ‹

___ › UR �

‹

__ › TS , RS � UT,

_ UR �

_ TS

Prove: �URT � �STR

Complete the proof.

Statements Reasons

1. ‹

__ › RS �

‹

__ › UT 1.

2. �SRT � �UTR 2.

3. ‹

___ › UR �

‹

__ › TS 3.

4. 4. Alt Int. � Thm.

5. �RUT � �RST 5. Third � Thm.

6. RS � UT 6.

7. 7. Def. � segments

8. _

UR � _

TS 8.

9. 9. Reflex. Prop of �

10. �URT � �STR 10.

Ready to Go On? Quiz continued

4ASECTION

33°

A

C

B (3x)°

L

N

M

5t + 4

7t – 1

29

U T

SR


Name Date Class


Name Date Class

Ready to Go On? Enrichment

Exploring Exterior AnglesFor Exercises 1–4, find the angle measures.

1. m�ABD

2. m�BDC

3. m�BCD

4. m�BCE

5. What is the sum of the measures of the exterior angles of the triangle?

For Exercises 6–9, find the angle measures.

6. m�1 7. m�2

8. m�3 9. m�4


For Exercises 11–17, find the indicated values.

11. x �

12. m�QSR

13. m�QSU

14. m�QRS 15. m�SRT

16. m�SQR 17. m�PQR


19. Make a conjecture about the sum of the measures of the exterior angles of a triangle.

4A

118°

47°

E C

BA

D

14

3

38°

2

(15x – 7)°

(7x – 3)°

(x2 – 13)°U S

QP

R

T

SECTION

Name Date Class



Using SSS and SAS to Prove Triangles Congruent_ JK � _

ML and _

JK � _

ML . Use SAS to explain why �JKM � �LMK.

It is given that _

JK � _

ML . This means that segment JK

is to segment ML. Mark this information on the figure.

It is given that _

JK � _


is to segment ML. Mark this information on the figure.

Since _

JK � _

ML , you know that � � �LMK because of the

Theorem.

By the Reflexive Property of Congruence, you know that _

MK � .

Therefore, � � � by .

Proving Triangles CongruentGiven:

_ AB �

_ BC , _

DB bisects �ABC. Prove: �ABD � �CBD

It is given that _

AB � _

BC and _

DB bisects �ABC. Mark this information on the figure.

Since _

DB bisects �ABC, you know that � � �

because of the definition of an .

Enter this information in Step 2 of the proof.

By the Reflexive Property of Congruence, you know that � .Enter this information in Step 3 of the proof.

Therefore, you know that �ABD � �CBD by . Enter this information in Step 4 of the proof.

Statements Reasons

1. _

AB � _

BC , _

DB bisects �ABC 1. Given

2. 2.

3. 3.

4. �ABD � �CBD 4.

Ready to Go On? Skills Intervention4-4 Triangle Congruence: SSS and SAS4B

SECTION

Vocabulary

triangle rigidity included angle

C

BD

A

LM

KJ


Name Date Class

Engineers often use triangles in designing structures because of their rigidity.

The figure shows a radio tower supported by cables of equal length. M is the midpoint of LN. Use SSS to explain why �PML � �PMN.


1. Why do you think a radio tower needs to be supported by cables?

2. Why do the cables form triangles with the tower and the ground?

3. The problem asks you to “Use SSS to explain why �PML � �PMN. When you explain something in Geometry, you must essentially write a paragraph proof. For every statement you make about the situation, you must also provide a

.

Make a PlanThe problem gives you information about the triangles that are formed by the tower, the cables, and the ground. Mark the figure with the given information as you answer each of the questions.

4. The sentence “The figure shows a tower supported by cables of equal length,”

tells you that � PN, and therefore, � .

5. The phrase “M is the midpoint of LN,” tells you that � .

6. The segment is congruent to itself.

SolveWrite a paragraph using the information you found in Exercises 4–6. Include justifications in your paragraph.

7. It is given that � , so � by the definition of

segments. By of a midpoint, � . By the

Property of Congruence, � . Therefore, �PML � �PMN by .

Look Back

8. To use the SSS Theorem to prove triangle congruence, 3 sides of one triangle

must be congruent to sides of a second triangle.

9. Have you proven that three sides of �PML are congruent to three sides of

�PMN? How?

Ready to Go On? Problem Solving Intervention4-4 Triangle Congruence: SSS and SAS4B

SECTION

L M N

P

Name Date Class


Find this vocabulary word in Lesson 4-5 and the Multilingual Glossary.

Applying HL CongruenceDetermine if you can use the HL Congruence Theorem to prove the triangles congruent. Explain.

A. �QPR and �SRP According to the diagram, �QPR and �SRP are

triangles that share leg .

� by the Reflexive Property of Congruence. Is any information given to you about the hypotenuse of the right triangles?

This conclusion be proven by HL. You need to know that the

of the triangles are .

B. �CDE and �CBE

According to the diagram, �CDE and �CBE are

triangles that share hypotenuse .

� by the Reflexive Property of Congruence.

It is given that � , therefore � � � by HL.

Using AAS to Prove Triangles CongruentGiven: �J � �L,

_ JK � _

ML Prove: �JKM � �LMK

Mark the given information on the figure.Since it is given that

_ JK � _

ML , you know that � � � .

Because of the Property of Congruence, you know that � .

Therefore, you know that � � � because of AAS.Complete the flow-chart.

4BSECTION

Vocabulary

included side

Ready to Go On? Skills Intervention4-5 Triangle Congruence: ASA, AAS, and HL

RP

SQ

E

DB

C

– –

�J � �LGiven

Given

� �1. 2.AAS

2.

�4.

�3.

�

3.

LM

KJ

_JK ��

_ML


Name Date Class

Ready to Go On? Skills Intervention4-6 Triangle Congruence: CPCTC4B

SECTION


Proving Corresponding Parts CongruentGiven: B is the midpoint of

_ AD ; _

AE � _

CD Prove:

_ BE �

_ BC

Mark the given information on the figure: B is the midpoint of _

AD and _

AE � _

CD .

Fill the given information into Step 1 and Step 3 of the flow-chart proof below.

Since B is the midpoint of _

AD , you know that � , because

of the definition of a .

Fill this information into Step 2 of the proof.

Since _

AE � _

CD , you know that � � �D and �E � �

because of the Angles Theorem.

Fill this information into Step 4 of your proof.

Therefore, �ABE � � by and � by CPCTC.

Fill this information into Steps 5 and 6 of your proof.

Complete the flow-chart:

Vocabulary

CPCTC

DE

CAB

� �4.

4.

�

� � �

Given

1.

2.

Given

3.

�2.

�6.

5.

5.

6.

� � �

Name Date Class



Positioning a Figure in the Coordinate PlanePosition a right triangle with legs of 7 units and 2 units in the coordinate plane.

Use the origin as the vertex of the right angle.

Count spaces to the right to find a second vertex.

Count 2 units from the origin to find the third vertex.

Connect the vertices to form a right triangle. Label the vertices with their coordinates.

Assigning Coordinates to VerticesPosition square LMNO in the coordinate plane and give the coordinates of each vertex.

Use the origin as one vertex of the square. Label it L.

Draw another vertex on the x-axis, to the right of origin. Label this vertex M(a, 0).

Move the same distance up from the origin on the y-axis and label this vertex O(0, a).

Describe where to place vertex N.

What are the coordinates of this vertex?

Connect the vertices to form a square.

Writing a Coordinate ProofUse the square LMNO you drew above to prove that

_ LN � _

MO .

Complete and use the distance formula: d � ��

( � x 1 ) 2 � ( y 2 � ) 2

Substitute the coordinates of L and N into the distance formula to find LN. Simplify.

LN � ��

( x 2 � x 1 ) 2 � ( y 2 � y 1 ) 2 � ��

(a � ) 2 � ( � 0 ) 2 �

Substitute the coordinates of M and O into the distance formula to find MO. Simplify.

MO � ��

( x 2 � x 1 ) 2 � ( y 2 � y 1 ) 2 � ��

(0 � ) 2 � (a � ) 2 �

Does LN � MO ?

So, � _

MO because of the definition of congruent segments.

Ready to Go On? Skills Intervention4-7 Introduction to Coordinate Proof4B

SECTION

Vocabulary

coordinate proof

x

y

2

2 4 6


Finding the Measure of an AngleFind m�L.

Look at the diagram. What type of triangle is �JKL?

From the Isosceles Triangle Theorem, you know that

�L � � . Therefore, m� � m� .

m�L � m�K Substitute the given values and solve to find x.

7x � 4 �

7x � 16 � Add 12 to both sides.

16 � Subtract 7x from both sides.

x � Divide both sides by 2.

Substitute the value of x into m�L and simplify.

m�L � 7x � 4 � 7( ) � 4 � � 4 �

Using Coordinate ProofGiven: Isosceles �JKL has coordinates J(�2a, 0), K(2a, 0)

and L(0, 4b). M is the midpoint of _

JL , N is the midpoint of

_ KL , and O is the midpoint of

_ JK .

Prove: �MNO is isosceles.

Use the Midpoint Formula M � � x 1 + x 2 ______

2 ,

y 1 + y 2 ______

2 � to find the coordinates of M, N, and O.

Coordinates of M Coordinates of N Coordinates of O

M � � � 0 _________ 2 , � 4b ________

2 �

� (�a, )

N � � 0 � _________ 2 , � 0 ________ 2 � � (a, )

O � � � 2a __________ 2 , � 0 _______

2 �

� (0, )

Draw �MNO on the diagram above.

Substitute the coordinates into the Distance Formula and simplify to find OM and ON.

OM � ��

( x 2 � x 1 ) 2 � ( y 2 � y 1 ) 2 � ��

( � 0 ) 2 � (2b � 0 ) 2 � ��

ON � ��

( x 2 � x 1 ) 2 � ( y 2 � y 1 ) 2 � ��

(a � 0 ) 2 � ( � 0 ) 2 � ��

Does OM � ON ? Since OM � ON, by definition, � _

ON .

Therefore, �MNO is an triangle.


Name Date Class

Ready to Go On? Skills Intervention4-8 Isosceles and Equilateral Triangles4B

SECTION

Vocabulary

legs of an isosceles triangle vertex angle base base angle

(7x + 4)x ° (9x – 12)°––

L K

J

––

J(–2JJ a, 0) K(2KK a, 0)

L(0, 4b)

Name Date Class


4-4 Triangle Congruence SSS and SAS 1. The figure shows the logo used for a department store. Given

that _

KI bisects �HKJ and _

KH � _

KJ , use SAS to explain why �KIH � �KIJ.

2. Given: _

UV � _

TW , _

UV � _

TW Prove : �VUW � �TWU

4-5 Triangle Congruence ASA, AAS, and HLDetermine if you can use the HL Congruence Theorem to prove the triangles congruent. If not, tell what else you need to know.

3. �ABD and �CDB 4. �NMO and �PMO

5. Use AAS to prove the triangles congruent. Given: K is the midpoint of

_ OM , _

ON � _

LM Prove: �LMK � �NOK

Ready to Go On? Quiz4B

SECTION

Statements Reasons

1. 1.

2. 2.

3. 3.

4. 4. WT

VU

H

I

J

K

M

PN

O

== ==

� �4.

4.

�

� � �

Given2.

Given

3.

.1 �2.

5.

�5.

CD

BA

M

N

L

O KK

4-6 Triangle Congruence CPCTC

6. Given _

TU � _

RS , _

TU � _

RS Prove:

_ QS �

_ QT

4-7 Introduction to Coordinate ProofPosition each figure in the coordinate plane.

7. a square with length 5 units 8. a right triangle with legs 5 units in length.

9. Assign coordinates to each vertex and write a coordinate proof Given: rectangle WXYZ Prove: WX � YZ

4-8 Isosceles and Equilateral TrianglesFind each angle measure.

10. m�Q (x + 4)°

(2x – 17)°––

Z Y

Q 11. m�E

48°==

––

C D

A

B

E

12. Given: Isosceles triangle LMN has coordinates L(0, 2b), M(2a, 0), and N(0, �2b). X is the midpoint of

_ LM and Y is the midpoint of

_ NM .

Prove: �XMY is isosceles.


Name Date Class


4BSECTION

Statements Reasons

1. 1.

2. 2.

3. 3.

4. 4.

5. 5.

T

S

U

R

Q

T

2–2 O–2

2

222222 2O2 2O–2 2O2 2O2 O

2

O 2–2 O–2

2

222222 2O2 2O–2 2O2 2O2 O

2

O

2–2 O–2

2

222222 2O2 2O–2 2O2 2O2 O

2

O

2–2 O–2

2

222222 2O2 2O–2 2O2 2O2 O

2

O

Name Date Class


Trying Triangles

1. In the figure at the right, X is the midpoint of _

AB . Write a paragraph to explain whether or not �BXM � �AXN.

2. In the figure at the right, _

ML � _

NO , and m�MOL � (2x � 2)�. Find �NOM.

3. Figure ABCD has coordinates A(2, 5), B (5, 1), C (1, �2) and D (�2, 2). m�A � m�B � m�C � m�D. What type of figure is ABCD? Does AC � BD? Explain how you got your answers.

4. What kind of triangle is formed by the lines y � 9x � 32, x � y � �2, and x � 9y � �32? Explain your answer.

5. In the figure at right, _

CD � _

BA . Is this enough information to show that �BDC � �DBA? Explain your reasoning.

Ready to Go On? Enrichment4B

SECTION

A

M

N

B X6y –y 5y 2 – 5

y 2 – 8

4y +y 3

OL

NM(15x + 24y)°

(5x – 4y)°

–=

–

=

B A

DC

x

y

2 4O

2

–2–2–4

–4

4

AAA

B

C

D

22O

x

y

2 4O

2

–2–2–4

–4

4

OOO



3-5 Slopes of Lines Use the slope formula to determine the slope of each line.

1. ‹

__ › AD

� 1 __ 2

2. ‹

__ › AB

7 __ 2

3. ‹

__ › AC

2 __ 4

or 1 __ 2

4. ‹

__ › DB

� 10 ___ 4 or � 5 __ 2

Find the slope of the line through the given points.

5. R(4, 7) and S(�2, 0) 6. C(0, �4) and D(5, 9)

7 __ 6

13 ___ 5

7. H(3, 5) and I(�4, 2) 8. S(�6, 1) and T(3, �6)

3 __ 7

� 7 __ 9

Graph each pair of lines and use their slopes to determine if they are parallel, perpendicular, or neither.

9. ‹

___ › CD and

‹

__ › AB for A(�1, 0), B(1, 5), 10.

‹

__ › LM and

‹

___ › MN for L(�3, 2), M(�1, 5),

C(4, 5), and D(�2, 4) N(2, 3), and P(1, �5)

2 4–2–4 O

2

–2

4

–4

22224224 2 4O24– 2 4O–2–4 2 4O24 2 4O24 O

2222

4

–4

O

2 4–2–4 O

2

–2

4

–4

22224224 2 4O24– 2 4O–2–4 2 4O24 2 4O24 O

2222

4

–4

O

Neither Perpendicular

11. ‹

__ › PR and

‹

__ › PS for P(2, �1), Q(2, 1), 12.

‹

___ › GH and

‹

__ › FJ for F(�3, 2), G(�2, 5)

R(�3, 1), and S(�2, �2) H(2, 4), and J(2, 1)

4–2–4 O

2

–2

4

–4

2 4–2–4 O

2

–2

4

–4

Neither Parallel


SECTION

x

y

2 4 6–2 O

2

–2

6

–4 B

D

C

A

––

4444

029-040_Ch3_RTGO_GEO_12738.indd 38 5/25/06 4:28:25 PM

3-6 Lines in the Coordinate PlaneWrite the equation of each line in the given form.

13. the line through (�3, �1) and (3, �3) in slope-intercept form y � � 1 __ 3 x � 2

14. the line through (6, �2) with slope � 3 __ 4 in point-slope form y � 2 � � 3 __

4 (x � 6)

15. the line with y-intercept �3 through the point (2, 5) in point-slope form y � 5 � 4(x � 2)

16. the line with x-intercept �4 and y-intercept 2 in slope-intercept form y � 1 __

2 x � 2

Graph each line.

17. y � 3x � 1 18. y � 1 � 3 __ 5

(x � 2) 19. y � �5

2 4–2–4

2

4

–4

2 4–2–4

2

4

–4

2 4–2–4 O–2

4

–4

2 4–2–4 O

2

–2

4

–4

22224224 2 4O24– 2 4O–2–4 2 4O24 2 4O24 O

2222

4

–4

O

Write the equation of each line.

20. 21. 22.

x

y

2–2–4 O

2

–2

4

–4

x

y

2–2–4 O

2

–2

4

x

y

2 4–2–4 O

2

–2

4

–4

x � 4 y � 3__

5 x � 3

y � �3

Determine whether the lines are parallel, intersect, or coincide.

23. y � � 1 __ 5

x � 2 24. 2x � 3y � 9 25. y � 5x � 3

x � 5y � 10 y � 2 __ 3

x � 1 y � 5x � 1

Coincide Intersect Parallel



3BSECTION

4444

029-040_Ch3_RTGO_GEO_12738.indd 39 10/13/05 9:45:37 AM


Slopes and Lengths of Segments

Quadrilateral ABCD has vertices A(�5, 3), B(�1, 4), C(5, �3) and D(�4, �1).

1. Sketch and label the quadrilateral using the grid at the right.

2. Find the slopes of _

AC and _

BD . � 3 __

5 and 5 __

3

3. How are the segments related? They are perpendicular.

Quadrilateral PQRS has vertices P(2, 3), Q(2, �2), R(�2, �5), S(�2, 0). Use the information to answer the following questions:

4. Sketch and label the quadrilateral using the grid at the right.

Find the length of each segment.

5. Find PQ. 6. Find QR.

5 units 5 units

7. Find RS. 8. Find PS.

5 units 5 units

9. What can you conclude about the side lengths of the quadrilateral? They are congruent.

10. What is the slope of _

PR ? 11. What is the slope of _

QS ?

2 � 1 __ 2

12. What can you conclude about the diagonals of the quadrilateral? They are perpendicular.

13. Is the quadrilateral a square? Explain your answer. No; the sides do not meet at

right angles.

14. A triangle has vertices L(2, 8), M(5, 9), and N(4, 2). Write a paragraph proof to show that triangle LMN is a right triangle.Sample answer: Using the slope formula; the slope of

_ LM is 1 __

3 and the slope of

_ LN is �3. The product of the slopes is �1, so by the Perpendicular Lines Theorem,

the segments are perpendicular. Perpendicular lines form right angles, and by

definition, a right triangle is a triangle that has one right angle.


SECTION

2 4–2–4 O

2

–2

4

–4

2 4–2–4 O

2

–2

4

–4

22224224 2 4O24– 2 4O–2–4 2 4O24 2 4O24 O

2222

4

–4

O

44444444

444444444444

4

444 2222

22 44

44

4444

4444

44444444

–4–444

222222222222 222222222222

029-040_Ch3_RTGO_GEO_12738.indd 40 10/13/05 9:45:39 AM


Classifying Triangles by Angle Measures

A. �PQS

A right angle has a measure of 90� .

Since �QPS is a right angle, �PQS is a right triangle.

B. �PRQ

First, find m�QRP. Since �QRP and �SRP form a linear pair, the angles are

supplementary . To find m�QRP, subtract 54� from 180� .

m�QRP � 180� � 54� � 126�

What kind of angle is �QRP ? Obtuse . So, �PRQ is an obtuse triangle.

C. �PRS

First, find m�RPS.

Since �RPS and �RPQ form a right angle, the angles are

complementary .

To find m�RPS, subtract 23� from 90� .

m�RPS � 90� � 23� � 67�

What kind of angle is �RPS? Acute .

What kind of angles are �SRP and �PSR ? Acute

So, �PRS is an acute triangle.

Classifying Triangles by Side Lengths

A. �JKL

How many sides are congruent in �JKL? 2

What kind of triangle is �JKL? Isosceles

B. �KML

Find KL. 9 How many sides are congruent, or have the same measure, in

�KML? None So, what kind of triangle is �KML? Scalene

4AReady to Go On? Skills Intervention4-1 Classifying Triangles

Vocabulary

acute triangle equiangular triangle right triangle obtuse triangle

equilateral triangle isosceles triangle scalene triangle

Q

P

R

S59°

54°23°

31°

K

JM

L7 4

98


SECTION



Complementary angles have a sum of 90�.

A roofer is making repairs on the roof of a house. In order to be safe, he sets his ladder so that it makes a 15� angle with the house. What angle will his ladder make with the ground?


1. What angle does the ladder make with the house? 15�

2. What angle does the house form with the ground? 90�

3. What kind of triangle is �ABC ? Right

Mark the figure with the information given in the problem.

Make a Plan

4. The acute angles of a right triangle are complementary, so the sum of the

measures of the acute angles equals 90� .

5. Complete: m� B � m�C � 90�

6. Write an equation by substituting the known angle measures. 15� � m� C � 90�

Solve

7. Solve the equation you wrote in Exercise 6:

15� � m�C � 90�

� 15� � 15� Subtract 15 from both sides to isolate the variable.

m�C � 75�

8. What angle does the roofer’s ladder make with the ground? 75�

Look BackYou can check your work in two ways.

9. What is the sum of the angles in a triangle? 180�

10. From your answer in Exercise 9, you know that m�A � m� B � m� C � 180� .

11. Substitute the angle measures and check your work. 90� � 15� � 75� � 180�

Does your answer check? Yes

12. To check using a second method, substitute your solution from Exercise 8

into the equation you wrote in Exercise 6: 15� � 75� � 90�

Does your answer check? Yes

4AReady to Go On? Problem Intervention4-2 Angle Relationships in Triangles

B

C Aground

ladder house


SECTION

15°



Finding Angle Measures in TrianglesFind m�C.

By the Triangle Sum Theorem, the sum of the angle

measures in a triangle is 180� .

In this triangle, m�A � m� B � m� C � 180�.

108� � 24� � m�C � 180� Substitute known measures.

132� � m�C � 180� Add.

� 132� � 132� Subtract to isolate the variable.

m�C � 48� Solve.

Finding Angle Measures in Right TrianglesOne of the acute angles in a right triangle measures 37.9�. What is the measure of the other acute angle?

Let the acute angles be �T and �U, with m�T � 37.9�.Since the measures of the acute angles in a right triangle are complementary,

m�T � m�U � 90� . Substitute 37.9� for m�T and solve for m�U.

37.9� � m�U � 90 �

� 37.9� � 37.9� Subtract to isolate the variable.

m�U � 52.1 � Solve.

Applying the Exterior Angles TheoremFind m�Q.

Using the Exterior Angles Theorem, m� P � m� Q � m�PRS. Substitute the given angle measures into the equation and solve for x.

44 � (5x � 3) � 8x � 5

47 � 5x � 8x � 5 Add.

5x � 8x � 42 Subtract 47 from both sides.

�3x � �42 Subtract 8x from both sides.

x � 14 Divide both sides by �3.

Substitute the value of x into (5x � 3) to find m�Q: (5x � 3) � (5)( 14 ) � 3 � 73�

4AReady To Go On? Skills Intervention4-2 Angle Relationships in Triangles

Vocabulary

Auxiliary line corollary interior exterior

interior angle exterior angle remote interior angle

108° 24°24°AA BB

CC

U

T

(8(8xx + 5)°+ 5)°xx (5(5xx + 3)°+ 3)°xx

44°44°

RRSS QQ

PP

4


SECTION


Ready To Go On? Skills Intervention4-3 Congruent Triangles


Naming Congruent Corresponding PartsGiven �BCD � �PQR. Identify the congruent corresponding parts to �B and

_ BD .

In a congruence statement, vertices are written in corresponding order .

�B corresponds with � P , so �B � � P .

_

BD corresponds with _

PR , so _

BD � _

PR .

Using Corresponding Parts of Congruent TrianglesGiven �DEF � �WXY.

A. Find the value of m.

�D corresponds with � W , so �D � � W .

Since �D � � W , m�D � m� W .

Substitute values for the angle measures D and W. Solve to find the value of m.

87 � 5m � 2

�2 �2 Subtract 2 from both sides.

85 � 5m

85 ___ 5 � 5m _____ 5 Divide both sides by 5.

17 � m Solve for m.

B. Find DE.

First find the value of x. _

XY corresponds with _

EF , so _

XY � _

EF and XY � EF . XY � EF Substitute values for XY and EF and solve for x.

3x � 7 � 26

�7 �7 Add 7 to both sides.

3x � 33

3x ___ 3 � 33 _____ 3 Divide both sides by 3.

x � 11 Solve for x.

Substitute the value of x into DE and simplify. DE � 2x � 9 � 2( 11 )� 9 � 13

4A

Vocabulary

corresponding angles corresponding sides congruent polygons

DD

EE FF26

WW

YYXX

2x – 9 87°

(5m + 2)°

3x – 7


SECTION



4-1 Classifying TrianglesClassify each triangle by its angle measures.

1. �QPR Acute

2. �SRQ Obtuse

3. �TRQ Right

Classify each triangle by its side lengths.

4. �QNM Scalene

5. �MPQ Isosceles

6. �NLM Equilateral

4-2 Angle Relationships in TrianglesFind each angle measure.

7. m�GFC 149� 8. m�BAC 44�

(8x + 19)°(12x – 7)°

26°

G F

C

D

(15x + 14)°(9x – 28)°

A C

B

D

9. A high school baseball team is designing a pennant with the school logo. The pennant is an isosceles triangle and the measure of the vertex angle is 46�.

Find the measure of the base angles. 67�

Ready to Go On? Quiz4A

SECTION

60° 60°

30°

30°

T

P SR

Q

8

8

= =8

L P Q N

M

46°



4-3 Congruent TrianglesGiven �MNO � �GHI. Identify the congruent corresponding parts.

10. _

MO � _

GI 11. _

GH � _

MN

12. �N � �H 13. �G � �M

Given �ABC � �LMN. Find each value.

14. LM 34

15. x 11

16. Given: ‹

__ › RS �

‹

__ › UT , ‹

___ › UR �

‹

__ › TS , RS � UT,

_ UR �

_ TS

Prove: �URT � �STR

Complete the proof.

Statements Reasons

1. ‹

__ › RS �

‹

__ › UT 1. Given

2. �SRT � �UTR 2. Alt. Int. � Thm

3. ‹

___ › UR �

‹

__ › TS 3. Given

4. �URT � �RTS 4. Alt Int. � Thm.

5. �RUT � �RST 5. Third � Thm.

6. RS � UT 6. Given

7. _

RS � _

UT 7. Def. � segments

8. _

UR � _

TS 8. Given

9. _

RT � _

RT 9. Reflex. Prop of �

10. �URT � �STR 10. Def. of � �


4ASECTION

33°

A

C

B (3x)°

L

N

M

5t + 4

7t – 1

29

U T

SR

s




Ready to Go On? Enrichment

Exploring Exterior AnglesFor Exercises 1–4, find the angle measures.

1. m�ABD 133�

2. m�BDC 62�

3. m�BCD 71�

4. m�BCE 109�

5. What is the sum of the measures of the exterior angles of the triangle? 360�

For Exercises 6–9, find the angle measures.

6. m�1 7. m�2

52� 142�

8. m�3 9. m�4

90� 128�


For Exercises 11–17, find the indicated values.

11. x � 7

12. m�QSR 46�

13. m�QSU 134�

14. m�QRS 15. m�SRT

36� 144�

16. m�SQR 17. m�PQR

98� 82�


19. Make a conjecture about the sum of the measures of the exterior angles of a triangle.

Sample answer: The sum of the exterior angle measures of a triangle

is always 360�.

4A

118°

47°

E C

BA

D

14

3

38°

2

(15x – 7)°

(7x – 3)°

(x2 – 13)°U S

QP

R

T

SECTION




Using SSS and SAS to Prove Triangles Congruent_ JK � _

ML and _

JK � _

ML . Use SAS to explain why �JKM � �LMK.

It is given that _

JK � _


is parallel to segment ML. Mark this information on the figure.

It is given that _

JK � _


is congruent to segment ML. Mark this information on the figure.

Since _

JK � _

ML , you know that � JKM � �LMK because of the

Alternate Interior Angles Theorem.


MK � _

KM .

Therefore, � JKM � � LMK by SAS .

Proving Triangles CongruentGiven:

_ AB �

_ BC , _

DB bisects �ABC. Prove: �ABD � �CBD

It is given that _

AB � _

BC and _

DB bisects �ABC. Mark this information on the figure.

Since _

DB bisects �ABC, you know that � ABD � � CBD

because of the definition of an angle bisector .

Enter this information in Step 2 of the proof.


DB � _

DB .Enter this information in Step 3 of the proof.

Therefore, you know that �ABD � �CBD by SAS . Enter this information in Step 4 of the proof.

Statements Reasons

1. _

AB � _

BC , _

DB bisects �ABC 1. Given

2. �ABD � �CBD 2. Def. of � bisector3.

_ DB �

_ DB 3. Reflex. Prop. of �

4. �ABD � �CBD 4. SAS

Ready to Go On? Skills Intervention4-4 Triangle Congruence: SSS and SAS4B

SECTION

Vocabulary

triangle rigidity included angle

C

BD

A

LM

KJ



Engineers often use triangles in designing structures because of their rigidity.

The figure shows a radio tower supported by cables of equal length. M is the midpoint of LN. Use SSS to explain why �PML � �PMN.


1. Why do you think a radio tower needs to be supported by cables?

Sample answer: To protect it from the wind.

2. Why do the cables form triangles with the tower and the ground?

Sample answer: Because of triangle rigidity

3. The problem asks you to “Use SSS to explain why �PML � �PMN. When you explain something in Geometry, you must essentially write a paragraph proof. For every statement you make about the situation, you must also provide a

justification .

Make a PlanThe problem gives you information about the triangles that are formed by the tower, the cables, and the ground. Mark the figure with the given information as you answer each of the questions.

4. The sentence “The figure shows a tower supported by cables of equal length,”

tells you that PL � PN, and therefore, _

PL � _

PN .

5. The phrase “M is the midpoint of LN,” tells you that _

ML � _

MN .

6. The segment _

PM is congruent to itself.

SolveWrite a paragraph using the information you found in Exercises 4–6. Include justifications in your paragraph.

7. It is given that PL � PN , so _

PL � _

PN by the definition of congruent

segments. By definition of a midpoint, _

ML � _

MN . By the Reflexive

Property of Congruence, _

PM � _

PM . Therefore, �PML � �PMN by SSS .

Look Back

8. To use the SSS Theorem to prove triangle congruence, 3 sides of one triangle

must be congruent to 3 sides of a second triangle.

9. Have you proven that three sides of �PML are congruent to three sides of

�PMN? How? Yes, PM � PM , PL � PN , ML � MN

Ready to Go On? Problem Solving Intervention4-4 Triangle Congruence: SSS and SAS4B

SECTION

L M N

P

041-056_CH04_RTGO_GEO_12738.indd 49 5/25/06 4:29:22 PM




Applying HL CongruenceDetermine if you can use the HL Congruence Theorem to prove the triangles congruent. Explain.

A. QPR and SRP According to the diagram, QPR and SRP are righttriangles that share leg

_PR ._

PR_RP by the Reflexive Property of Congruence.

Is any information given to you about the hypotenuse of the right triangles? No

This conclusion cannot be proven by HL. You need to know that the hypotenuse of the triangles are congruent .

B. CDE and CBE

According to the diagram, CDE and CBE are righttriangles that share hypotenuse

_CE ._

CE_CE by the Reflexive Property of Congruence.

It is given that

_BE

_DE , therefore CDE CBE by HL.

Using AAS to Prove Triangles CongruentGiven: J L,

_JK

_ML

Prove: JKM LMK

Mark the given information on the figure.Since it is given that

_JK

_ML , you know that JKM LMK .

Because of the Reflexive Property of Congruence, you know that

_MK

_KM .

Therefore, you know that JKM LMK because of AAS.Complete the flow-chart.

4BSECTION

Vocabulary

included side

Ready to Go On? Skills Intervention4-5 Triangle Congruence: ASA, AAS, and HL

RP

SQ

E

DB

C

– –

J LGiven

Given

1. 2.AAS

2.

4.

3.

3.

LM

KJ

LMKJKM LMKJKM

_MK

_KM

Alt. Int. Thm

Refl. Prop. Of

_JK

_ML



Ready to Go On? Skills Intervention4-6 Triangle Congruence: CPCTC4B

SECTION


Proving Corresponding Parts CongruentGiven: B is the midpoint of

_ AD ; _

AE � _

CD Prove:

_ BE �

_ BC

Mark the given information on the figure: B is the midpoint of _

AD and _

AE � _

CD .

Fill the given information into Step 1 and Step 3 of the flow-chart proof below.

Since B is the midpoint of _

AD , you know that _

AB � _

DB , because

of the definition of a midpoint .

Fill this information into Step 2 of the proof.

Since _

AE � _

CD , you know that � A � �D and �E � � C

because of the Alternate Interior Angles Theorem.

Fill this information into Step 4 of your proof.

Therefore, �ABE � � DBC by AAS and _

BE � _

BC by CPCTC.

Fill this information into Steps 5 and 6 of your proof.

Complete the flow-chart:

Vocabulary

CPCTC

DE

CAB

� �4.

4.

�

� � �

Given

1.

2.

Given

3.

�2.

�6.

5.

5.

6.

� � �

_

DB Def. of a Midpt.

_

AE � _

CD

_

AB B is the midpoint of AD

A D

E C

Alt. Int. � Thm

ABE DBC

AAS

_

BE

CPCTC

_

BC




Positioning a Figure in the Coordinate PlanePosition a right triangle with legs of 7 units and 2 units in the coordinate plane.

Use the origin as the vertex of the right angle.

Count 7 spaces to the right to find a second vertex.

Count 2 units up from the origin to find the third vertex.

Connect the vertices to form a right triangle. Label the vertices with their coordinates.

Assigning Coordinates to VerticesPosition square LMNO in the coordinate plane and give the coordinates of each vertex.

Use the origin as one vertex of the square. Label it L.

Draw another vertex on the x-axis, to the right of origin. Label this vertex M(a, 0).

Move the same distance up from the origin on the y-axis and label this vertex O(0, a).

Describe where to place vertex N. Sample answer: Vertex N will be a units up from vertex M and a units to the right of vertex O.

What are the coordinates of this vertex? The coordinates are (a, a).

Connect the vertices to form a square.

Writing a Coordinate ProofUse the square LMNO you drew above to prove that

_ LN � _

MO .

Complete and use the distance formula: d � ��

( x 2 � x 1 ) 2 � ( y 2 � y 1 ) 2

Substitute the coordinates of L and N into the distance formula to find LN. Simplify.

LN � ��

( x 2 � x 1 ) 2 � ( y 2 � y 1 ) 2 � ��

(a � 0 ) 2 � ( a � 0 ) 2 � a ��

2

Substitute the coordinates of M and O into the distance formula to find MO. Simplify.

MO � ��

( x 2 � x 1 ) 2 � ( y 2 � y 1 ) 2 � ��

(0 � a ) 2 � (a � 0 ) 2 � a ��

2

Does LN � MO ? Yes

So, _

LN � _

MO because of the definition of congruent segments.

Ready to Go On? Skills Intervention4-7 Introduction to Coordinate Proof4B

SECTION

Vocabulary

coordinate proof

x

y

2

2 4 6



Finding the Measure of an AngleFind m�L.

Look at the diagram. What type of triangle is �JKL? Isosceles

From the Isosceles Triangle Theorem, you know that

�L � � K . Therefore, m� L � m� K .

m�L � m�K Substitute the given values and solve to find x.

7x � 4 � 9x � 127x � 16 � 9x Add 12 to both sides.

16 � 2x Subtract 7x from both sides.

x � 8 Divide both sides by 2.

Substitute the value of x into m�L and simplify.

m�L � 7x � 4 � 7( 8 ) � 4 � 56 � 4 � 60�

Using Coordinate ProofGiven: Isosceles �JKL has coordinates J(�2a, 0), K(2a, 0)

and L(0, 4b). M is the midpoint of _

JL , N is the midpoint of

_ KL , and O is the midpoint of

_ JK .

Prove: �MNO is isosceles.

Use the Midpoint Formula M � � x 1 + x 2 ______

2 ,

y 1 + y 2 ______

2 � to find the coordinates of M, N, and O.

Coordinates of M Coordinates of N Coordinates of O

M � � �2a � 0 _________ 2 , 0 � 4b ________

2 �

� (�a, 2b )

N � � 0 � 2a _________ 2 , 4b � 0 ________ 2 � � (a, 2b )

O � � �2a � 2a __________ 2 , 0 � 0 _______

2 �

� (0, 0 )

Draw �MNO on the diagram above.

Substitute the coordinates into the Distance Formula and simplify to find OM and ON.

OM � ��

( x 2 � x 1 ) 2 � ( y 2 � y 1 ) 2 � ��

( �a � 0 ) 2 � (2b � 0 ) 2 � �� a 2 � 4 b 2

ON � ��

( x 2 � x 1 ) 2 � ( y 2 � y 1 ) 2 � ��

(a � 0 ) 2 � ( 2b � 0 ) 2 � �� a 2 � 4 b 2

Does OM � ON ? Yes Since OM � ON, by definition, _

OM � _

ON .

Therefore, �MNO is an isosceles triangle.


Ready to Go On? Skills Intervention4-8 Isosceles and Equilateral Triangles4B

SECTION

Vocabulary

legs of an isosceles triangle vertex angle base base angle

(7x + 4)x ° (9x – 12)°––

L K

J

––

J(–2JJ a, 0) K(2KK a, 0)

L(0, 4b)

O (0, 0)

N(NN a, 2b)M(–MM a, 2b)




4-4 Triangle Congruence SSS and SAS 1. The figure shows the logo used for a department store. Given

that _

KI bisects �HKJ and _

KH � _

KJ , use SAS to explain why �KIH � �KIJ.

Sample answer: Since _

KI bisects �HKJ, �HKI � �JKI ;it is given that

_ KH �

_ KJ : _

KI � _

KI by the Reflexive Property of Congruence. So �KIH � �KIJ by SAS.

2. Given: _

UV � _

TW , _

UV � _

TW Prove : �VUW � �TWU

4-5 Triangle Congruence ASA, AAS, and HLDetermine if you can use the HL Congruence Theorem to prove the triangles congruent. If not, tell what else you need to know.

3. �ABD and �CDB 4. �NMO and �PMO

Not enough information. Yes

Need to know that

AD � BC or _

AB � _

CD .

5. Use AAS to prove the triangles congruent. Given: K is the midpoint of

_ OM , _

ON � _

LM Prove: �LMK � �NOK


SECTION

Statements Reasons

1. _

UV � _

TW , _

UV � _

TW 1. Given2. �VUW � �TWU 2. Alt. Int. � Thm3. _

UW � _

WU 3. Reflex. Prop. Of �4. �VUW � �TWU 4. SASWT

VU

H

I

J

K

M

PN

O

== ==

� �4.

4.

�

� � �

Given2.

Given

3.

.1 �2.

5.

�5.

CD

BA

M

N

L

O KK

K is mdpt. of _

OM _

OK _

MK

_

ON � _

LM AAS

Alt. Int. � Thm

�LMK �NOK

L NM O

Def. of mdpt.


4-6 Triangle Congruence CPCTC

6. Given _

TU � _

RS , _

TU � _

RS Prove:

_ QS �

_ QT

4-7 Introduction to Coordinate ProofPosition each figure in the coordinate plane. Check student’s graphs 7. a square with length 5 units 8. a right triangle with legs

5 units in length.

9. Assign coordinates to each vertex and write a coordinate proof Given: rectangle WXYZ Prove: WX � YZ

Check student’s graph and verify distance formula calculations.

4-8 Isosceles and Equilateral TrianglesFind each angle measure.

10. m�Q (x + 4)°

(2x – 17)°––

Z Y

Q 11. m�E

48°==

––

C D

A

B

E

46� 12�

12. Given: Isosceles triangle LMN has coordinates L(0, 2b), M(2a, 0), and N(0, �2b). X is the midpoint of

_ LM and Y is the midpoint of

_ NM .

Prove: �XMY is isosceles.

coordinate of X (a, b), coordinate of Y (a, �b),

MX � MY � ��

a 2 � b 2 Copyright © by Holt, Rinehart and Winston. 55 Holt GeometryAll rights reserved.


4BSECTION

Statements Reasons

1. TU � RS; TU � RS 1. Given2. �T � �S 2. Alt. Int. � Thm3. �TQU � �SQR 3. Vert � Thm4. �TQU � �SQR 4. AAS5. QS � QT 5. CPCTC

T

S

U

R

Q

T

2–2 O–2

2

222222 2O2 2O–2 2O2 2O2 O

2

O 2–2 O–2

2

222222 2O2 2O–2 2O2 2O2 O

2

O

2–2 O–2

2

222222 2O2 2O–2 2O2 2O2 O

2

O

2–2 O–2

2

222222 2O2 2O–2 2O2 2O2 O

2

O



Trying Triangles

1. In the figure at the right, X is the midpoint of _

AB . Write a paragraph to explain whether or not �BXM � �AXN.

The triangles are not congruent. y � 6,

XA � XB � 31, XM � 27 but XN � 28.

2. In the figure at the right, _

ML � _

NO , and m�MOL � (2x � 2)�. Find �NOM.

38�

3. Figure ABCD has coordinates A(2, 5), B (5, 1), C (1, �2) and D (�2, 2). m�A � m�B � m�C � m�D. What type of figure is ABCD? Does AC � BD? Explain how you got your answers.

AB � BC � CD � AD � 5;

AC � BD � 5 ��

2 ; Since all angles are

90� and all sides have the same

lengths, then ABCD is a square.

4. What kind of triangle is formed by the lines y � 9x � 32, x � y � �2, and x � 9y � �32? Explain your answer.

This is an isosceles triangle. The

points of intersection of the lines are

(4, 4), (3, �5), and (�5, 3). The

lengths of the sides of the triangles

are ��

82 , ��

82 , and ��

128 � 8 ��

2 .

5. In the figure at right, _

CD � _

BA . Is this enough information to show that �BDC � �DBA? Explain your reasoning.

Sample answer: Yes. Given CD � BA, �CDB � �ABD by Alt. Int.

� Thm. �CDB � �CBD by Isoc. � Thm. �ABD � �CBD by subst.

�ABD � �ADB by Isoc. � Thm. �ADB � �CDB by subst. BD � BD

by Reflex. Prop. of �. �BDC � �DBA by ASA.


SECTION

A

M

N

B X6y –y 5y 2 – 5

y 2 – 8

4y +y 3

OL

NM(15x + 24y)°

(5x – 4y)°

–=

–

=

B A

DC

x

y

2 4O

2

–2–2–4

–4

4

AAAAAAA

BBB

CCC

DDD

2222OO

x

y

2 4O

2

–2–2–4

–4

4

OOOOOO



Applying the Perpendicular Bisector Theorem and Its ConverseFind each measure.

A. DF

Since CD � CE, and � � _

DE , ‹

___ › CF is the perpendicular

bisector of _

DE by the Converse of the

Perpendicular Bisector Theorem. Therefore, DF � FE because of the definition of a

segment bisector .

Substitute 3 for FE.

DF � 3

B. VU

TU � VU because of the Perpendicular Bisector Theorem. Substitute the given measures for TU and VU and solve for x.

12x � 3 � 14x � 5

12x � 12x � 3 � 5 � 14x � 5 � 5 � 12x

8 � 2 x

4 � x

Substitute the value of x to find VU. 14x � 5

� 14( 4 ) � 5

� 51

Applying the Angle Bisector TheoremFind QR.

QR � RS because of the Angle Bisector Theorem.Substitute 46 for RS.

QR � 46

Ready to Go On? Skills Intervention5-1 Perpendicular and Angle Bisectors

Vocabulary

equidistant focus

SECTION


5A

C

D

E

F5.8

5.83

�

TU

V

W�

12x + 3

14x – 5

P

QR

S

4627°27°

�


Documents

SECTION Ready to Go On? Skills Intervention 4A 4-1 ... angles have a sum of 90 . ... interior angle exterior angle remote interior angle ... Ready to Go On? Skills Intervention