1. Section 6 – 6 Use Proportionality Theorem

2. Theorems Triangle Proportionality Theorem – If a line parallel to one side of a triangle intersects the other two sides, then it divided the two sides proportionally. Converse of the Triangle Proportionality Theorem – If a line divides two sides of a triangle proportionally, then it is parallel to the third side.

3. Theorems Theorem 6.6 – If three parallel lines intersect two transversals, then they divide the transversals proportionally. Theorem 6.7 – If a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides.

4. Example 1 In the diagram, RS || PN, MS = 15, SN = 20, and RP = 12. What is the length of MR? MS = MR SN RP MR 12 Triangle Proportionality Theorem 15 = 20 Cross Multiply 20MR = 180 S N M P 12 15 R 20 20 20 MR = 9

5. Example 2 In the diagram, ABD = CBD. Use the given side lengths to find the length of DC. Because BD is an angle bisector of ABC, we can apply Theorem 6.6. DA = DC BA BC D 32 24 40 xC B A Angle bisector divides opposite side proportionally. Substitute 40 – x = x 24 32 24x =32(40 – x) 24x = 1280 – 32x ~ 56x = 1280 x = 22.9

6. Homework Section 6-6 Page 400 – 403 3 – 6, 8 – 11, 13 – 16, 30 – 33