Similarity Exploration

• Use a protractor and a ruler to draw two noncongruent triangles so that each triangle has a 400 angle and a 600 angle.

• What can you determine about these figures? Why?

Proving Triangles SimilarStudents will be able to prove triangles similar using the AA, SSS, SAS similarity theorem.

Angle-Angle Similarity Postulate (AA~ Post.)

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

If JKL XYZ and KJL YXZ, then JKL XYZ.

J

K

LX

Y

Z

Proportionalitya. Write the similarity statement.b.Write the statement of proportionality.c.Find mTEC.d.Find ET and BE.

E

T

B

C

W

20

340

790

12

3

€

a. ΔBTW ~ ΔETC

b. ETBT

=TCTW

=CEWB

c. ∠B ≅∠TEC, so m∠TEC = 790

d. 3

12=ET20

5 = ET

State if the triangles in each pair are similar. If so, state how you know they are similar and complete the similarity

statement.

€

HTS

State if the triangles in each pair are similar. If so, state how you know they are similar and complete the similarity

statement.

€

not similar

State if the triangles in each pair are similar. If so, state how you know they are similar and complete the similarity

statement.

€

TUV

State if the triangles in each pair are similar. If so, state how you know they are similar and complete the similarity

statement.

Not similar

Find the missing length. The triangles are similar.

x = 9€

ABC ~ ΔFED

Find the missing length. The triangles are similar.

x = 9

Find JU. The triangles are similar.

x = 24

Find PW. The triangles are similar.

x = 11

Given: Prove: WVX ~ ZYX

€

VW YZ

W

V

X Z

Y

€

1. VW || YZ2. ∠V ≅∠Y3. ∠VXW ≅∠YXZ4. ΔWVX ~ ΔZYX

€

1. Given2. Alternate Interior Angles Theorem3. Vertical Angles4. A - A Postulate

Statements Reasons

Given: ABC is a right triangle, AD is an altitude

Prove: ABC DAC

A

B

C

D

€

1. ΔABC is a right triangle, AD is an altitude,

m∠BAC = 900

2. ∠ADC is right angle

3. m∠ADC = 900

4. ∠C = ∠C5. ΔABC ~ ΔDAC

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1. Given

2. Definition of altitude3. Definition of right angle4. Reflexive Prop5. A - A postulate

Statements Reasons

Theorem 8.2 Side-Side-Side (SSS) Similarity Theorem

If the corresponding sides of two triangles are proportional, then the triangles are similar.If , then ABC PQR.

€

ABPQ

=BCQR

=CARP

A

B

CP

Q

R

Theorem 8.3 Side-Angle-Side (SAS) Similarity Theorem

If an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar.

If X M and ,then XYZ MNP.

€

ZXPM

=XYMN

X

Y

Z

M

N

P