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7- 3 Triangle Similarity: AA, SSS, and SAS When two triangles have 2 pairs of congruent angles, the third pair must be congruent too. The lengths of the triangles are in proportion. Angle-Angle (AA~) Similarity Postulate—If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. ~ http://www.google.com/imgres?q=aa+similarity+postulate&num=10& How is this AA ~ ? Side-Angle-Side (SAS~) Similarity Theorem—if an angle of one triangle is congruent to an angle of another triangle, and the sides including these angles are in proportion, then the triangles are similar. If < V is congruent to <L, and UV = WV, then triangle UVW ~ triangle KL KL ML Make sure you put the “~” mark when using similarity !!! Side-Side-Side (SSS~) Similarity Theorem --If all corresponding sides of two triangles are in proportion, then the triangles are similar If AB = BC = CA, then triangle ABC ~ triangle HJK HJ JK KH Properties of Similarity 1. Reflexive Property of Similarity ABC ~ ABC 2. Symmetric Property of Similarity If ABC ~ DEF, then DEF ~ ABC 3. Transitive Property of Similarity If ABC ~ DEF and DEF ~ XYZ, then ABC ~ XYZ PROOF Given: BC ll to DE Prove: Triangle ABC ~ Triangle ADE Statement Reason 1. BC ll DE 1. Given 2. <A is congruent to <A 2. Reflexive 3. <B is congruent to <D 3. Corresponding <’s 4. <C is congruent to < E 4. Corresponding <’s 5. Triangle ABC ~ Triangle ADE 5. AA Similarity Postulate