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Objectives

• Prove that two triangles are similar using AA, SAS, and SSS

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Proving Two Triangles Similar with Shortcuts

• Instead of using the definition of similarity to prove that two triangles are congruent (all corresponding angles are congruent and all corresponding sides are proportional), you can use three shortcuts:– Angle-Angle (AA)– Side-Angle-Side (SAS)– Side-Side-Side (SSS)

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Angle-Angle (AA) Similarity Postulate

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

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AA Example

Explain why the triangles are similar and write a similarity statement.

∠R ≅ ∠V (Given)

∠RSW ≅ ∠VSB (vertical angles are congruent)

ΔRSW ≅ ΔVSB (AA)

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Side-Angle-Side (SAS) Similarity Theorem

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

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CB

A

H

J

G

ΔABC ~ ΔGJH

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SAS Example

Explain why the two triangles are similar and write a similarity statement.

∠Q ≅ ∠X since they are right angles

The two sides that include the right angles are proportional

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6

4

3

By SAS, ΔPRQ ~ΔZYX

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Side-Side-Side (SSS) Similarity Theorem

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

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CB

A

H

J

G

ΔABC ~ ΔGJH

105

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SSS Example

Explain why the two triangles are similar and write the similarity statement.

Since all sides of the two triangles are proportional, by SSS, ΔABC ~ ΔEFG

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EF

AB

GF

CB

EG

AC