PostulateIf two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

If . . .∠S ≅ ∠M and ∠R ≅ ∠L

Then . . .△SRT ∼ △MLP

Postulate 9-1 Angle-Angle Similarity (AA =) Postulate

R

S T M P

L

TheoremIf an angle of one triangle is congruent to an angle of a second triangle, and the sides that include the two angles are proportional, then the triangles are similar.

If . . .ABQR = AC

QS and ∠A ≅ ∠Q

Then . . .△ABC ∼ △QRS

Theorem 9-1 Side-Angle-Side Similarity (SAS =) Theorem

A

B C

Q

R S

TEKS (7)(B) Apply the Angle-Angle criterion to verify similar triangles and apply the proportionality of the corresponding sides to solve problems.

TEKS (1)(E) Create and use representations to organize, record, and communicate mathematical ideas.

Additional TEKS (1)(F), (1)(G), (8)(A)

TEKS FOCUSIndirect measurement – Indirect measurement is a way of measuring things that are difficult to measure directly.

Representation – a way to display or describe information. You can use a representation to present mathematical ideas and data.

VOCABULARY

ESSENTIAL UNDERSTANDING

You can show that two triangles are similar when you know the relationships between only two or three pairs of corresponding parts.

Sometimes you can use similar triangles to find lengths that cannot be measured easily using a ruler or other measuring device.

You will prove Theorem 9-1 in Exercise 21.

9-3 Proving Triangles Similar

392 Lesson 9-3 Proving Triangles Similar

TheoremIf the corresponding sides of two triangles are proportional, then the triangles are similar.

If . . .ABQR = AC

QS = BCRS

Then . . .△ABC ∼ △QRS

Theorem 9-2 Side-Side-Side Similarity (SSS =) Theorem

A

B C

Q

R S You will prove Theorem 9-2 in Exercise 22.

Problem 2

Problem 1

Using the AA = Postulate

Are the two triangles similar? How do you know?

A △RSW and △VSB

∠R ≅ ∠V because both angles measure 45°. ∠RSW ≅ ∠VSB because vertical angles are congruent. So △RSW ∼ △VSB by the AA ∼ Postulate.

B △JKL and △PQR

∠L ≅ ∠R because both angles measure 70°. By the Triangle Angle-Sum Theorem, m∠K = 180 - 30 - 70 = 80 and m∠P = 180 - 85 - 70 = 25. Only one pair of angles is congruent. So △JKL and △PQR are not similar.

W V

BS

R

45!

45!

Q

R P

85!

70!

70!

K

J L30!

Verifying Triangle SimilarityAre the triangles similar? If so, write a similarity statement for the triangles.

A

Use the side lengths to identify corresponding sides. Then set up ratios for each pair of corresponding sides.

Shortest sides STXV = 6

9 = 23

Longest sides USWX = 10

15 = 23

Remaining sides TUVW = 8

12 = 23

All three ratios are equal, so corresponding sides are proportional. △STU ∼ △XVW by the SSS ∼ Theorem.

TEKS Process Standard (1)(E)

X

V WS

TU 8

69 15

1210

X

V WS

TU 8

69 15

1210

What do you need to show that the triangles are similar? To use the AA ∼ Postulate, you need to prove that two pairs of angles are congruent.

continued on next page ▶

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Problem 3

continuedProblem 2

B

∠K ≅ ∠K by the Reflexive Property of Congruence.

KLKM = 8

10 = 45 and KP

KN = 1215 = 4

5.

So △KLP ∼ △KMN by the SAS ∼ Theorem.

K P N

LM2

8

312

L

K

8

∠K is the included angle betweentwo known sides in each triangle.

M

8 ! 2 " 10

P12 K N12 ! 3 " 15

How can you make it easier to identify corresponding sides and angles?Sketch and label two separate triangles.

Proving Triangles Similar

Given: FG ≅ GH , JK ≅ KL, ∠F ≅ ∠J

Prove: △FGH ∼ △JKL

Find two pairs of corresponding congruent angles and use the AA ∼ Postulate to prove the triangles are similar.

You need to show that the triangles are similar.

The triangles are isosceles, so the base angles are congruent.

Statements Reasons

1) FG ≅ GH , JK ≅ KL 1) Given

2) △FGH is isosceles. △JKL is isosceles.

2) Def. of an isosceles △

3) ∠F ≅ ∠H, ∠J ≅ ∠L 3) Base ⦞ of an isosceles △ are ≅.

4) ∠F ≅ ∠J 4) Given

5) ∠H ≅ ∠J 5) Transitive Property of ≅

6) ∠H ≅ ∠L 6) Transitive Property of ≅

7) △FGH ∼ △JKL 7) AA ∼ Postulate

Proof

TEKS Process Standard (1)(G)

G

F HJ

K

L

394 Lesson 9-3 Proving Triangles Similar

34 ft34 ft34 ft34 ft

x ftx ftx ftx ft

6 ft6 ft6 ft6 ft

5.5 ft5.5 ft5.5 ft5.5 ft

SSSS

JJJJ

VVVV

HHHH

TTTT

PRACTICE and APPLICATION EXERCISES

ONLINE

HO M E W O R

K

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

Use Representations to Communicate Mathematical Ideas (1)(E) Determine whether the triangles are similar. If so, write a similarity statement and name the postulate or theorem you used. If not, explain.

1. 2. 3. F

G

H

J

K

U N R

AY

S25!

35!

35! 110!

R P12 24

16

8

S

Q

T

Problem 4

Applying Proportionality to Indirect Measurement

Rock Climbing Before rock climbing, Darius wants to know how high he will climb. He places a mirror on the ground and walks backward until he can see the top of the cliff in the mirror. What is the height of the cliff?

△HTV ∼ △JSV AA ∼ Postulate

HTJS = TV

SV Corresponding sides of ∼ triangles are proportional.

5.5x = 6

34 Substitute.

187 = 6x Cross Products Property

31.2 ≈ x Solve for x.

The cliff is about 31 ft high.

Before solving for x, verify that the triangles are similar. △HTV ∼ △ JSV by the AA ∼ Postulate because ∠T ≅ ∠S and ∠HVT ≅ ∠JVS .

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4. Given: ∠ABC ≅ ∠ACD 5. Given: PR = 2NP, 6. Given: PQ # QT, ST # TQ,

Prove: △ABC ∼ △ACD PQ = 2MP PQST = QR

TV Prove: △MNP ∼ △QRP Prove: △VKR is isosceles.

Explain why the triangles are similar. Then find the distance represented by x.

7. 8.

9. Analyze Mathematical Relationships (1)(F) At a certain time of day, a 1.8-m-tall person standing next to the Washington Monument casts a 0.7-m shadow. At the same time, the Washington Monument casts a 65.8-m shadow. How tall is the Washington Monument?

10. a. Are two isosceles triangles always similar? Explain.

b. Are two right isosceles triangles always similar? Explain.

11. Apply Mathematics (1)(A) A 2-ft vertical post casts a 16-in. shadow at the same time that a nearby cell phone tower casts a 120-ft shadow. How tall is the cell phone tower?

12. Justify Mathematical Arguments (1)(G) Does any line that intersects two sides of a triangle and is parallel to the third side of the triangle form two similar triangles? Justify your reasoning.

13. Draw any △ABC with m∠C = 30. Use a straightedge and compass to construct △LKJ so that △LKJ ∼ △ABC.

14. Explain Mathematical Ideas (1)(G) In the diagram at the right, △PMN ∼ △SRW . MQ and RT are altitudes. The scale factor of △PMN to △SRW is 4 : 3. What is the ratio of MQ to RT ? Explain how you know.

15. △ABC has vertices A(0, 0), B(2, 4), and C(4, 2). △RST has vertices R(0, 3), S(-1, 5), and T(-2, 4). Prove that △ABC ∼ △RST .

Proof Proof Proof

A

C

D B

M

NP

Q

R P

Q V R T

S

K

x90 ft

120 ft135 ft

Mirror

5 ft 6 in.

4 ft 10 ft

x

PW R

ST

NM

Q

Proof

396 Lesson 9-3 Proving Triangles Similar

TEXAS Test Practice

23. Complete the statement △ABC ∼ ? . By which postulate or theorem are the triangles similar?

A. △AKN ; SSS ∼ C. △ANK ; SAS ∼ B. △AKN ; SAS ∼ D. △ANK ; AA ∼

24. ∠1 and ∠2 are alternate interior angles formed by two parallel lines and a transversal. If m∠2 = 68, what is m∠1?

F. 22 G. 68 H. 112 J. 122

25. The length of a rectangle is twice its width. If the perimeter of the rectangle is 72 in., what is the length of the rectangle?

A. 12 in. B. 18 in. C. 24 in. D. 36 in.

26. Graph A(2, 4), B(4, 6), C(6, 4), and D(4, 2). What type of polygon is ABCD? Justify your answer.

A 12

8

2

3

BN

C K

For each pair of similar triangles, find the value of x.

16. 17. 18.

19. Write a proof of the following: Any two nonvertical parallel lines have equal slopes.

Given: Nonvertical lines /1 and /2, /1 } /2, EF and BC are # to the x-axis

Prove: BCAC = EF

DF 20. Use the diagram in Exercise 19. Prove: Any two nonvertical lines with equal

slopes are parallel.

21. Write a paragraph proof to prove the Side-Angle-Side Similarity Theorem (Theorem 9-1).

Given: ABQR = AC

QS, ∠A ≅ ∠Q

Prove: △ABC ∼ △QRS

22. Write a paragraph proof to prove the Side-Side-Side Similarity Theorem (Theorem 9-2).

Given: ABQR = AC

QS = BCRS

Prove: △ABC ∼ △QRS

2x

89

5

x

2x ! 4

x " 639

24

10

x ! 2

x ! 14

x

Proof

A C D F

E

x

y

B

O

!1

!2

ProofA

B C

Q

R S

Proof

ProofA

B C

Q

R S

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