Lesson 13.2 Similar Triangles pp. 542-547

Objectives: 1. To identify and illustrate the basic similarity postulate for triangles. 2. To illustrate, state, and prove the SSS Similarity Theorem. 3. To identify the SAS Similarity Theorem. 4. To apply the similarity postulate and theorems to pairs of triangles with similarity.

What does ABC ~ XYZ mean?

If ABC ~ XYZ then A  X, B  Y, C  Z, and AC XZ = BC YZ AB XY

Postulate 13.1 AA Similarity Postulate. If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar.

The AA Similarity Postulate allows us to say two triangles are similar if only two pairs of corresponding angles are known to be congruent.

Theorem 13.1 SSS Similarity Theorem. If the three sides of one triangle are proportional to the corresponding three sides of another triangle, then the triangles are similar.

Theorem 13.2 SAS Similarity Theorem. If two sides of one triangle are proportional to the corresponding two sides of another triangle and the included angles between the sides are congruent, then the triangles are similar.

Theorem 13.3 Similarity of triangles is an equivalence relation.

Definition An equivalence relation is a relation that is reflexive, symmetric, and transitive.

Homework pp. 545-547

►A. Exercises State whether the lengths (in units) below could be sides of similar triangles. 1. 3, 8, 7, and 12, 32, 28

►A. Exercises State whether the lengths (in units) below could be sides of similar triangles. 3. 12, 15, 9, and 4, 5, 3

►A. Exercises State whether the lengths (in units) below could be sides of similar triangles. 5. 7, 4, 9, and 14, 10, 18

►A. Exercises Which pair of triangles are similar? Why? 7.

►A. Exercises Which pair of triangles are similar? Why? 9. 16 8 12 6 4

►A. Exercises Which pair of triangles are similar? Why? 11.

13. Given: WXYZ is a parallelogram Prove: WXY ~ YZW ►B. Exercises Prove the following statements. 13. Given: WXYZ is a parallelogram Prove: WXY ~ YZW W X Z Y

►B. Exercises 13. Statements Reason 1. WXYZ is a parallel. 1. Given 2. ZY||WX; ZW||YX 2. Def. of Parallel. 3. ZYW  YWX 3. Parallel Post. ZWY  XYW 4. WXY  YZW 4. AA

■ Cumulative Review 21. What is a relation that is reflexive, symmetric, and transitive?

■ Cumulative Review 22. List three symbols that represent equivalence relations.

■ Cumulative Review 23. Does the set of rotations with a given center P form an equivalence relation?

■ Cumulative Review 24. Suppose two regions are related if they have the same area. Is this an equivalence relation? Why?

■ Cumulative Review 25. Suppose two solids with the same volume are related. Is this an equivalence relation? Why?