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6.4, 6.5 Name:________________ Prove Triangles Similar by AA, SSS and SAS Learning Target: By the end of today’s lesson we will be able to successfully use the AA, SSS or SAS Similarity Postulates. Angle-Angle (AA) Similarity Postulate: If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar. JKL ~ XYZ Example 1: Determine whether the triangles are similar. If they are, write a similarity statement. Explain your reasoning. a.) b.) c.) Example 2: Show that the two triangles are similar. a.) RTV and RQS b.) LMN and NOP Side-Side-Side (SSS) Similarity Theorem: If the corresponding side lengths of two triangles are proportional then the triangles are similar. If AB BC CA = = , then ∆ABC ~ ∆RST. RS ST TR c.) BCD and EFD Example 3: Is either ∆DEF or ∆GHJ similar to ∆ABC? Example 4: Find the value of x that makes ∆ABC ~ ∆DEF. 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 ______________, then the triangles are similar. If ∠X ≅ ∠M , and ZX XY = , then ∆XYZ ∼ ∆MNP. PM MN Example 5: A lifeguard is standing beside the lifeguard chair on a beach. The lifeguard is 6 feet 4 inches tall and casts a shadow that is 48 inches long. The chair casts a shadow that is 6 feet long. Are the triangles similar. If so, how tall is the chair? *************************************************************************************************************************************** Triangle Similarity Postulate and Theorems: AA Similarity Postulate: SSS Similarity Theorem: If ∠A ≅ ∠D and ∠B ≅ ∠E, then ∆ABC ~ ∆DEF. (If 2 angles of 1 triangle = 2 angles of another triangle they are similar) If AB BC AC = = , then ∆ABC ~ ∆DEF. DE EF DF (If all sides of 1 triangle proportional to all sides of another triangle they are similar) SAS Similarity Theorem: If ∠A ≅ ∠D and AB AC = , then ∆ABC ∼ ∆DEF. DE DF (If all 2sides of 1 triangle proportional to 2 sides of another triangle and the included angles are = then the triangles are similar)