Lesson 7-5 Theorems for Similar Triangles (page 263) Essential Question How can similar polygons be used to solve real life problems?

Way to Prove ANY Two Triangles Similar AA ~ Postulate

Theorem 7-1 SAS ~ Theorem If an angle of one triangle is congruent to an angle of another triangle and the sides including those angles are in proportion , then the triangles are similar. D Given: ∠A ≅ ∠D Prove: ∆ ABC ~ ∆ DEF A E F B C

Theorem 7-2 SSS ~ Theorem If the sides of two triangles are in proportion , then the triangles are similar. D Given: Prove: ∆ ABC ~ ∆ DEF A E F B C

Ways to Prove ANY Two Triangles Similar AA ~ Postulate SAS ~ Theorem SSS ~ Theorem

To prove two polygons similar, you may need to compare corresponding sides. A useful technique is to compare the longest sides, the shortest sides, and so on.

Perimeters of similar polygons are in the same ratio as the corresponding sides. By using similar triangles, you can prove that corresponding segments, such as diagonals, also altitudes, medians, etc. of similar polygons, also have this ratio .

∴ ∆PQR ~ ∆TSP by the SSS ~ Theorem Compare the longest sides: Example #1 Can the information given be used to prove ∆PQR ~ ∆TSP? If so, how? SP = 8, TS = 6, PT = 12, QR = 12, PQ = 9, RP= 18. Q ∴ ∆PQR ~ ∆TSP by the SSS ~ Theorem 12 9 18 P R 12 T 8 6 S Compare the longest sides: Compare the shortest sides: Compare the other sides:

SP = 7, TS = 6, PQ = 9, QR = 10.5, m∠S = 80º, m∠QPR + m∠QRP = 100º. Example #2 Can the information given be used to prove ∆PQR ~ ∆TSP? If so, how? SP = 7, TS = 6, PQ = 9, QR = 10.5, m∠S = 80º, m∠QPR + m∠QRP = 100º. Q ∴ ∆PQR ~ ∆TSP by the SAS ~ Theorem 80º 10.5 9 If m∠QPR + m∠QRP = 100º, then m ∠Q = 80º. m∠QPR + m∠QRP = 100º P R T 7 6 80º S Compare the longer sides: Compare the shorter sides:

How can similar polygons be used to solve real life problems? Classroom Exercises on pages 264 & 265 1 to 6 Assignment Written Exercises on pages 266 & 267 REQUIRED: 1 to 7 ALL numbers, 9, 14, 20 Prepare for Quiz on Lessons 7-4 & 7-5 How can similar polygons be used to solve real life problems?