30!

x !

80!

D

C

AB

PRACTICE and APPLICATION EXERCISES

ONLINE

HO M E W O R

K

For additional support whencompleting your homework, go to PearsonTEXAS.com.

1. Justify Mathematical Arguments (1)(G) Write a paragraph proof to prove the Triangle Angle-Sum Theorem (Theorem 3-11). Begin by drawing an auxiliary line through vertex T.

Given: △STU

Prove: m∠S + m∠T + m∠U = 180

Find the value of each variable.

2. 3. 4.

Find each missing angle measure.

5. 6. 7.

ProofS

U T

30!

80!

40!

x ! y ! z !

70! 30!

x !

y !

30!

c !

1 63!

60!2

13!

128.5!

3 4

45!

47!

Problem 4

Applying the Triangle Theorems

Multiple Choice When radar tracks an object, the reflection of signals off the ground can result in clutter. Clutter causes the receiver to confuse the real object with its reflection, called a ghost. At the right, there is a radar receiver at A, an airplane at B, and the airplane’s ghost at D. What is the value of x?

30 70

50 80

m∠A + m∠B = m∠BCD Triangle Exterior Angle Theorem

x + 30 = 80 Substitute.

x = 50 Subtract 30 from each side.

The value of x is 50. The correct answer is B.

Scan page for a Virtual Nerd™ tutorial video.

How can you apply your skills from Problem 3 here?Look at the diagram. Notice that you have a triangle and information about interior and exterior angles.

114 Lesson 3-5 Parallel Lines and Triangles

8. A ramp forms the angles shown at the right. What are the values of a and b?

9. Analyze Mathematical Relationships (1)(F) What is the measure of each angle of a triangle with three congruent angles? Explain.

10. A beach chair has different settings that change the angles formed by its parts. Suppose m∠2 = 71 and m∠3 = 43. Find m∠1.

Use the given information to find the unknown angle measures in the triangle.

11. The ratio of the angle measures of the acute angles in a right triangle is 1∶2.

12. The measure of one angle of a triangle is 40. The measures of the other two angles are in a ratio of 3∶4.

13. The measure of one angle of a triangle is 108. The measures of the other two angles are in a ratio of 1∶5.

14. Analyze Mathematical Relationships (1)(F) The angle measures of △RST are represented by 2x, x + 14, and x - 38. What are the angle measures of △RST ?

15. Prove the following theorem: The acute angles of a right triangle are complementary.

Given: △ABC with right angle C

Prove: ∠A and ∠B are complementary.

Find the values of the variables and the measures of the angles.

16. 17.

18. 19.

a! b! 72!

Proof

A

B

C

(2x ! 9)"

(2x # 4)"

x"

Q

P R

(8x ! 1)"

(4x # 7)"

C

A

B

b!

a!

c !

d ! 32!

55!

e!

G

F

H

Ex !

z !

y !

54! 52!

A D

B

C

115PearsonTEXAS.com

1

3

2

20. Prove the Triangle Exterior Angle Theorem (Theorem 3-12).

The measure of each exterior angle of a triangle equals the sum of the measures of its two remote interior angles.

Given: ∠1 is an exterior angle of the triangle.

Prove: m∠1 = m∠2 + m∠3

21. Without using the Triangle Angle-Sum Theorem as a reason, write a two-column proof to prove that the acute angles of a right triangle are complementary.

Given: △ABC with right angle ACB

Prove: ∠BAC and ∠ABC are complementary.

22. Explain Mathematical Ideas (1)(G) Two angles of a triangle measure 64 and 48. What is the measure of the largest exterior angle of the triangle? Explain.

23. Analyze Mathematical Relationships (1)(F) A right triangle has exterior angles at each of its acute angles with measures in the ratio 13∶14. Find the measures of the two acute angles of the right triangle.

24. In the figure at the right, CD # AB and CD bisects ∠ACB. Find m∠DBF .

25. If the remote interior angles of an exterior angle of a triangle are congruent, what can you conclude about the bisector of the exterior angle? Justify your answer.

Proof

1 4

2

3

CA

B ED

A

CF B

D (3x ! 2) "

(5x ! 20)"

TEXAS Test Practice

26. The measure of one angle of a triangle is 115. The other two angles are congruent. What is the measure of each of the congruent angles?

A. 32.5 B. 57.5 C. 65 D. 115

27. One statement in a proof is “∠1 and ∠2 are supplementary angles.” The next statement is “m∠1 + m∠2 = 180.” Which is the best justification for the second statement based on the first statement?

F. The sum of the measures of two right angles is 180.

G. Angles that form a linear pair are supplementary.

H. Definition of supplementary angles

J. The measure of a straight angle is 180.

28. △ABC has one obtuse angle, m∠A = 21, and ∠C is acute.

a. What is m∠B + m∠C? Explain.

b. What is the range of whole numbers for m∠C? Explain.

c. What is the range of whole numbers for m∠B? Explain.

116 Lesson 3-5 Parallel Lines and Triangles

2. Select Tools to Solve Problems (1)(C) Consider the following conjecture.

If two triangles have the same perimeter, then the triangles are congruent.

a. Select a real object that you can use to test the conjecture. Explain your choice.

b. Is the conjecture true? If not, make a new conjecture based on your results. Explain your reasoning.

3. Explain Mathematical Ideas (1)(G) At least how many triangle measurements must you know in order to guarantee that all triangles built with those measurements will be congruent? Explain your reasoning.

4. Given: IE ≅ GH, EF ≅ HF, 5. Given: WZ ≅ ZS ≅ SD ≅ DW

F is the midpoint of GI Prove: △WZD ≅ △SDZ

Prove: △EFI ≅ △HFG

What other information, if any, do you need to prove the two triangles congruent by SAS? Explain.

6. 7.

8. Evaluate Reasonableness (1)(B) You and a friend are cutting triangles out of felt for an art project. You want all the triangles to be congruent. Your friend tells you that each triangle should have two 5-in. sides and a 40° angle. If you follow this rule, will all your felt triangles be congruent? Explain.

Can you prove the triangles congruent? If so, write the congruence statement and name the postulate you would use. If not, write not enough information and tell what other information you would need.

9. 10. 11.

Proof Proof

EF

G

HI

W

D

S

Z

G

L T

N

M Q

TU

W

VS

R

A G

NR T

W

H

Y

D

P

K

T

J S

E VF

156 Lesson 4-2 Triangle Congruence by SSS and SAS

12. Use Representations to Communicate Mathematical Ideas (1)(E) Sierpinski’s triangle is a famous geometric pattern. To draw Sierpinski’s triangle, start with a single triangle and connect the midpoints of the sides to draw a smaller triangle. If you repeat this pattern over and over, you will form a figure like the one shown. This particular figure started with an isosceles triangle. Are the triangles outlined in red congruent? Explain.

13. Create Representations to Communicate Mathematical Ideas (1)(E) Use a straightedge to draw any triangle JKL. Then construct △MNP ≅ △JKL using the given postulate.

a. SSS b. SAS

14. Analyze Mathematical Relationships (1)(F) Suppose GH ≅ JK , HI ≅ KL, and ∠I ≅ ∠L. Is △GHI congruent to △JKL? Explain.

15. Given: FG } KL, FG ≅ KL

Prove: △FGK ≅ △KLF

Proof

F G

KL

TEXAS Test Practice

17. What additional information do you need to prove that △VWY ≅ △VWZ by SAS?

A. YW ≅ ZW C. ∠Y ≅ ∠Z

B. ∠WVY ≅ ∠WVZ D. VZ ≅ VY

18. The measures of two angles of a triangle are 43 and 38. What is the measure of the third angle?

F. 9 G. 81 H. 99 J. 100

19. Which method would you use to find the inverse of a conditional statement?

A. Negate the hypothesis only. C. Negate the conclusion only.

B. Switch the hypothesis and D. Negate both the hypothesis and the conclusion. the conclusion.

Y

Z

WV

16. Given: AB # CM, AB # DB, CM ≅ DB,

M is the midpoint of AB.

Prove: △AMC ≅ △MBD

DB

M

A

C

Proof

157PearsonTEXAS.com

PRACTICE and APPLICATION EXERCISESON

LINE

HO M E W O R

K

For additional support whencompleting your homework, go to PearsonTEXAS.com.

Determine whether the triangles must be congruent. If so, name the postulate or theorem that justifies your answer. If not, explain.

1. 2. 3.

4. Given: ∠FJG ≅ ∠HGJ, FG } JH 5. Given: PQ # QS, RS # SQ,

Prove: △FGJ ≅ △HJG T is the midpoint of PR

Prove: △PQT ≅ △RST

6. Evaluate Reasonableness (1)(B) While helping your family clean out the attic, you find the piece of paper shown at the right. The paper contains clues to locate a time capsule buried in your backyard. The maple tree is due east of the oak tree in your backyard. Will the clues always lead you to the correct spot? Explain.

7. Connect Mathematical Ideas (1)(F) Anita says that you can rewrite any proof that uses the AAS Theorem as a proof that uses the ASA Postulate. Do you agree with Anita? Explain.

8. Justify Mathematical Arguments (1)(G) Can you prove that the triangles at the right are congruent? Justify your answer.

9. Given: ∠N ≅ ∠P, MO ≅ QO 10. Given: ∠1 ≅ ∠2, and

Prove: △MON ≅ △QOP DH bisects ∠BDF

Prove: △BDH ≅ △FDH

P NO

M T

S

RU

V

Z Y

W

Proof Proof

F

H

G

J

P

Q

T S

R

Proof Proof

M

QP

N

O

F

D

BH

21

Scan page for a Virtual Nerd™ tutorial video.

161PearsonTEXAS.com

11. Given: AB } DC, AD } BC

Prove: △ABC ≅ △CDA

12. Create Representations to Communicate Mathematical Ideas (1)(E) Draw two noncongruent triangles that have two pairs of congruent angles and one pair of congruent sides.

13. Given AD } BC and AB } DC, name as many pairs of congruent triangles as you can.

14. Create Representations to Communicate Mathematical Ideas (1)(E) Use a straightedge to draw a triangle. Label it △JKL. Construct △MNP ≅ △JKL so that the triangles are congruent by ASA.

15. Prove the Angle-Angle-Side Theorem (Theorem 4-2). Use the diagram next to it on page 158.

16. In △RST at the right, RS = 5, RT = 9, and m∠T = 30. Show that there is no SSA congruence rule by constructing △UVW with UV = RS, UW = RT , and m∠W = m∠T , but with △UVW R △RST .

ProofBA

CD

A

E

D

B C

R

TS

59

30!

TEXAS Test Practice

17. Suppose RT ≅ ND and ∠R ≅ ∠N. What additional information do you need to prove that △RTJ ≅ △NDF by ASA?

A. ∠T ≅ ∠D C. ∠J ≅ ∠D

B. ∠J ≅ ∠F D. ∠T ≅ ∠F

18. You plan to make a 2 ft-by-3 ft rectangular poster of class trip photos. Each photo is a 4 in.-by-6 in. rectangle. If the photos do not overlap, what is the greatest number of photos you can fit on your poster?

F. 4 H. 32

G. 24 J. 36

19. Write the converse of the true conditional statement below. Then determine whether the converse is true or false.

If you are less than 18 years old, then you are too young to vote in the United States.

162 Lesson 4-3 Triangle Congruence by ASA and AAS

Problem 2

Writing a Proof Using the HL Theorem

Given: BE bisects AD at C, AB # BC, DE # EC, AB ≅ DE

Prove: △ABC ≅ △DEC

Proof

A

B

CE

D

BE bisects AD.Given

AB ⊥ BCDE ⊥ EC

Given

GivenAB ≅ DE

∠ABC and∠DEC areright ⦞.

Def. of ⊥ lines△ABC ≅ △DEC

HL Theorem

AC ≅ DC

Def. of bisector

△ ABC and △ DECare right .

Def. of right triangle

PRACTICE and APPLICATION EXERCISES

ONLINE

HO M E W O R

K

For additional support whencompleting your homework, go to PearsonTEXAS.com.

1. Justify Mathematical Arguments (1)(G) Copy the flow chart and complete the proof.

Given: PS ≅ PT , ∠PRS ≅ ∠PRT

Prove: △PRS ≅ △PRT

2. Study Exercise 1. Can you prove that △PRS ≅ △PRT without using the HL Theorem? Explain.

3. Explain Mathematical Ideas (1)(G) Complete the paragraph proof.

Given: ∠A and ∠D are right angles, AB ≅ DE

Prove: △ABE ≅ △DEB

Proof: It is given that ∠A and ∠D are right angles. So, a. ? by the definition of right triangles. b. ? , because of the Reflexive Property of Congruence. It is also given that c. ? . So, △ABE ≅ △DEB by d. ? .

RS

P

T

∠PRS and ∠PRT are ≅.

∠PRS and ∠PRTare supplementary.

△PRS and △PRTare right .

△PRS ≅ △PRT

e.

Given

⦞ that form a linearpair are supplementary.

b.

∠PRS and ∠PRTare right ⦞.

a.

PS ≅ PT

c.

PR ≅ PR

d.

E

B

A

D

Scan page for a Virtual Nerd™ tutorial video.

How can you get started? Identify the hypotenuse of each right triangle. Prove that the hypotenuses are congruent.

176 Lesson 4-6 Congruence in Right Triangles

4. Given: HV # GT , GH ≅ TV , I is the midpoint of HV

Prove: △IGH ≅ △ITV

5. Given: PM ≅ RJ , PT # TJ , RM # TJ , M is the midpoint of TJ

Prove: △PTM ≅ △RMJ

J

R

MT

P

Connect Mathematical Ideas (1)(F) For what values of x and y are the triangles congruent by HL?

6. 7.

8. Apply Mathematics (1)(A) △ABC and △PQR are right triangular sections of a fire escape, as shown. Is each story of the building the same height? Explain.

9. Connect Mathematical Ideas (1)(F) “Aha!” exclaims your classmate. “There must be an HA Theorem, sort of like the HL Theorem!” Is your classmate correct? Explain.

10. Given: △LNP is isosceles with base NP, MN # NL, QP # PL, ML ≅ QL

Prove: △MNL ≅ △QPL

MN P

Q

L

Create Representations to Communicate Mathematical Ideas (1)(E) Copy the triangle and construct a triangle congruent to it using the given method.

11. SAS 12. HL

13. ASA 14. SSS

Proof

G

H

T

VI

Proof

x x ! 3 y ! 13y x ! 5y " x

y ! 5

3y ! x

RA B

C

P Q

RA B

C

P Q

Proof

177PearsonTEXAS.com

15. Given: △GKE is isosceles with base GE, ∠L and ∠D are right angles, and K is the midpoint of LD.

Prove: LG ≅ DE

16. Given: LO bisects ∠MLN , OM # LM , ON # LN

Prove: △LMO ≅ △LNO

17. Justify Mathematical Arguments (1)(G) Are the triangles at the right congruent? Explain.

Analyze Mathematical Relationships (1)(F) For Exercises 18 and 19, use the figure at the right.

18. Given: BE # EA, BE # EC, △ABC is equilateral

Prove: △AEB ≅ △CEB

19. Given: △AEB ≅ △CEB, BE # EA, BE # EC

Can you prove that △ABC is equilateral? Explain.

Proof

KL

G E

D

Proof

N

O

M

L

A

B

D

E

135

5

13

FC

Proof

A CE

B

TEXAS Test Practice

20. You often walk your dog around the neighborhood. Based on the diagram at the right, which one of the following  statements about distances is true?

A. SH = LH C. SH 7 LH

B. PH = CH D. PH 6 CH

21. In equilateral △XYZ , name four pairs of congruent right triangles. Explain why they are congruent.

School (S)

Park (P)

Café (C )

Library (L)

Home (H)

X

ZR

SP Q

Y

178 Lesson 4-6 Congruence in Right Triangles

PRACTICE and APPLICATION EXERCISES

ONLINE

HO M E W O R

K

For additional support whencompleting your homework, go to PearsonTEXAS.com.

In each diagram, the red and blue triangles are congruent. Identify their common side or angle.

1. 2. 3.

Separate and redraw the indicated triangles. Identify any common sides or angles.

4. △PQS and △QPR 5. △ACB and △PRB 6. △JKL and △MLK

P

M

NL

K ED

FG

T

W

X

ZY

Q

S R

T

P P

B

CR

A K L

O

J M

Separating Overlapping Triangles

Given: CA ≅ CE , BA ≅ DE

Prove: BX ≅ DX

Statements Reasons

1) BA ≅ DE 1) Given

2) CA ≅ CE 2) Given

3) ∠CAE ≅ ∠CEA 3) Base ⦞ of an isosceles △ are ≅.

4) AE ≅ AE 4) Reflexive Property of ≅5) △BAE ≅ △DEA 5) SAS

6) ∠ABE ≅ ∠EDA 6) Corresp. parts of ≅ △s are ≅.

7) ∠BXA ≅ ∠DXE 7) Vertical angles are ≅.

8) △BXA ≅ △DXE 8) AAS

9) BX ≅ DX 9) Corresp. parts of ≅ △s are ≅.

Proof

TEKS Process Standard (1)(G)

C

B DX

A E

A

B DX

EE AA

B D

E

Problem 4

Which triangles are useful here?If △BXA ≅ △DXE, then BX ≅ DX because they are corresponding parts. If △BAE ≅ △DEA, you will have enough information to show △BXA ≅ △DXE.

Scan page for a Virtual Nerd™ tutorial video.

181PearsonTEXAS.com

7. Justify Mathematical Arguments (1)(G) Complete the flow proof.

Given: ∠T ≅ ∠R, PQ ≅ PV

Prove: ∠PQT ≅ ∠PVR

8. Given: RS ≅ UT , RT ≅ US

Prove: △RST ≅ △UTS

9. Given: QD ≅ UA, ∠QDA ≅ ∠UAD

Prove: △QDA ≅ △UAD

10. Given: ∠1 ≅ ∠2, ∠3 ≅ ∠4

Prove: △QET ≅ △QEU

11. Given: AD ≅ ED, D is the midpoint of BF

Prove: △ADC ≅ △EDG

12. Explain Mathematical Ideas (1)(G) In the diagram at the right, ∠V ≅ ∠S, VU ≅ ST, and PS ≅ QV. Which two triangles are congruent by SAS? Explain.

13. Identify a pair of overlapping congruent triangles in the diagram. Then use the given information to write a proof to show that the triangles are congruent.

Given: AC ≅ BC, ∠A ≅ ∠B

P

V Q

ST R

d.∠PQT ≅ ∠PVR

e.

∠T ≅ ∠R

a.

∠TPQ ≅ ∠RPV △TPQ ≅ △RPVb.

PQ ≅ PV

c.

Proof

S

M

T

W V

UR

Proof

Q U

A

R

D

Proof

T

34 2

1

U

EQ B

Proof

G

C

DB

A

F

E

P

W

V

R

STU

X

Q

A

D E

B

F

C

182 Lesson 4-7 Congruence in Overlapping Triangles

14. Apply Mathematics (1)(A) The figure at the right is part of a clothing design pattern, and it has the following relationships.

GC # AC

AB # BC

AB } DE } FG

m∠A = 50

△DEC is isosceles with base DC.

a. Find the measures of all the numbered angles in the figure.

b. Suppose AB ≅ FC. Name two congruent triangles and explain how you can prove them congruent.

15. Given: AC ≅ EC , CB ≅ CD

Prove: ∠A ≅ ∠E

16. Given: QT # PR, QT bisects PR, QT bisects ∠VQS

Prove: VQ ≅ SQ

17. Create Representations to Communicate Mathematical Ideas (1)(E) Draw a quadrilateral ABCD with AB } DC, AD } BC, and diagonals AC and DB intersecting at E. Label your diagram to indicate the parallel sides.

a. List all the pairs of congruent segments in your diagram.

b. Explain how you know that the segments you listed are congruent.

STEM

B

C

4

1 2 3 6

7

98

I 5

A D F

H JE

G

Proof

B

A

C

FD

E

Proof

P Q

T

V S

R

Proof

TEXAS Test Practice

18. According to the diagram at the right, which statement is true?

A. △DEH ≅ △GFH by AAS C. △DEF ≅ △GFE by AAS

B. △DEH ≅ △GFH by SAS D. △DEF ≅ △GFE by SAS

19. △ABC is isosceles with base AC. If m∠C = 37, what is m∠B?

F. 37 G. 74 H. 106 J. 143

20. Which word correctly completes the statement “All ? angles are congruent”?

A. adjacent B. supplementary C. right D. corresponding

21. In the figure, LJ } GK and M is the midpoint of LG.

a. Copy the diagram. Then mark your diagram with the given information.

b. Prove △LJM ≅ △GKM .

c. Can you prove that △LJM ≅ △GKM another way? Explain.

D

G

E

H

F

L

J

MK

G

183PearsonTEXAS.com