1. Objectives
Prove that two triangles are similar using AA, SAS, and SSS

2. 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)

3. Angle-Angle (AA) Similarity Postulate
If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

4. 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)

5. 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. G A 2 4 B C 3 J 6 H
ΔABC ~ ΔGJH

6. 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
By SAS, ΔPRQ ~ΔZYX

7. Side-Side-Side (SSS) Similarity Theorem
If the corresponding sides of two triangles are proportional, then the triangles are similar. G A 4 5 8 10 B C 6 J 12 H
ΔABC ~ ΔGJH

8. 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