7.1.3 Triangle Similarity: AA, SSS, and SAS Key Objectives • Prove certain triangles are similar by using AA, SSS, and SAS. • Use triangle similarity to solve problems. Theorems, Postulates, Corollaries, and Properties • Angle-Angle Similarity Postulate If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. • Side-Side-Side Similarity Theorem If the three sides of one triangle are proportional to the three corre- sponding sides of another triangle, then the triangles are similar. • Side-Angle-Side Similarity Theorem If two sides of one triangle are proportional to two sides of anoth- er triangle and their included angles are congruent, then the triangles are similar. • Reflexive Property of Similarity △ ABC ∼ △ ABC • Symmetric Property of Similarity If △ ABC ∼ △ DEF , then △ DEF ∼ △ ABC . • Transitive Property of Similarity If △ ABC ∼ △ DEF and △ DEF ∼ △ XYZ , then △ ABC ∼ △ XYZ . Example 1 Using the AA Similarity Postulate
If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
The reason why two triangles are similar is determined in this example. It is given that each triangle has a right angle. Angles B and D are congruent because they are both right angles, as given. According to the Vertical Angles Theorem, ∠ BCA ≅ ∠ DCE . Therefore, by Angle-Angle Similarity the two triangles are similar.
Example 2 Verifying Triangle Similarity
If three sides of one triangle are proportional to the three corresponding sides of another triangle, then the triangles are similar.
19
Made with FlippingBook - PDF hosting