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Understanding Similarity in Triangles: AA, SAS, and SSS Theorems Explained

This guide introduces methods to prove that two triangles are similar using Angle-Angle (AA), Side-Angle-Side (SAS), and Side-Side-Side (SSS) theorems. The AA postulate states that if two angles of one triangle are congruent to two angles of another, the triangles are similar. The SAS theorem elucidates that if one angle is congruent and the sides surrounding the angles are proportional, the triangles are also similar. Lastly, the SSS theorem confirms that proportionality of corresponding sides establishes the similarity of triangles.

jillian-adriano
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Understanding Similarity in Triangles: AA, SAS, and SSS Theorems Explained

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  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

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