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7.3 Triangle Similarity: AA, ASA, SSS Objectives: G.SRT.5: Use congruence and similarity criteria for triangles to solve problems and prove relationships in geometric figures. G.SRT.3: Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. G.SRT.2: Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar. For the Board: You will be able to prove triangles are similar by using AA, SSS, and SAS. Anticipatory Set: According to the definition of Similar Figures, to be similar all the angles of one figure must be congruent to all the angles of the other and all the sides must be proportional. This is more than is necessary for triangles. Angle-Angle (AA) Similarity Postulate If two angles of the triangle are congruent to two angles of another triangle, then the two A X triangles are similar. Given: <A <X, <B <Y Conclusion: ΔABC ~ ΔXYZ C B Z Y Side-Side-Side (SSS) Similarity Theorem If the lengths of the corresponding sides of two triangles are proportional, then the triangles are similar. A X AB BC AC Given: XY YZ XZ Conclusion: ΔABC ~ ΔXYZ C B Y Z Side-Angle-Side (SAS) Similarity Theorem If an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar. AB BC A Given: <B <Y, X XY YZ Conclusion: ΔABC ~ ΔXYZ B C Y Z Open the book to page 483 and read example 1. Example: In each of the following: a. Are the triangles similar? YES or NO b. If YES, what postulate or theorem is the reason? AA SAS SSS c. IF YES, write the similarity statement. 1. a. YES b. AA B <BCA <DCE by the Vertical Angle Theorem E c. ΔBCA ~ ΔDCE C A D 2. P 2 Q 3 U 3. 4.5 4.5 5.8 2 2.9 3 S a. YES 3/4.5 = 30/45 = 2/3 b. SSS c. ΔRPQ ~ ΔUTS 1 70° H F E T J K 70° 3 R D a. YES 2/1 = 2, 5.8/2.9 = 2 b. SAS c. ΔDEF ~ ΔHJK Complete Handout 7.3 Open the book to page 484 and read example 5. Example: Given: BE||CD Prove: ΔABE ~ ΔACD Proof: Statements Reasons 1. BE||CD 1. Given 2. <ABE <ACE 2. Corresponding Angles Post. 3. <A <A 3. Reflexive Property of 4. ΔABE ~ ΔACD 4. AA White Board Activity: Practice: Given: AC/EC = BC/DC Prove: Prove ΔABC ~ ΔEDC Proof: Statements Reasons 1. AC/ED = BC/DC 1. Given 2. <ACB <ECD 2. Vertical Angle Th. 3. ΔABC ~ ΔEDC 3. SAS A B E C D D A C B E Assessment: Collect and grade Handout 7.3 Independent Practice: Text: pg. 486 – 487 prob. 1 – 4, 7, 11 – 14 , 18. Hint: In #18 <1 <2 by Corresponding Angles of Similar Triangles are Congruent (CASTC). For a Grade: Text: pgs. 486 – 487 prob. 12, 14, 18.