1. Give the postulate or theorem that justifies why the triangles are similar. ANSWER AA Similarity Postulate 2. Solve = . 16 32 12 – x x ANSWER 8

Use ratios and proportions to solve geometric problems. Target Use ratios and proportions to solve geometric problems. You will… Use proportions with a triangle or parallel lines.

Vocabulary Triangle Proportionality Theorem 6.4 – If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally. Know… so you can say QS is parallel to UT RQ QU RS ST = Triangle Proportionality Converse Theorem 6.5 – If a line divides two sides of a triangle proportionally, then it is parallel to the third side.

Vocabulary Parallel Lines Proportionality Theorem 6.6 – If three parallel lines intersect two transversals, then they divide the transversals proportionally. VX XZ UW WY =

part 1 side 1 part 2 part 1 part 2 side 1 side 2 side 2 Vocabulary Angle Bisector Proportionality 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. part 1 Above the bisector AD DB CA CB = side 1 part 2 below the bisector part 1 part 2 side 1 side 2 = side 2

Find the length of a segment EXAMPLE 1 Find the length of a segment In the diagram, QS || UT , RS = 4, ST = 6, and QU = 9. What is the length of RQ ? SOLUTION RQ QU RS ST = Triangle Proportionality Theorem RQ 9 4 6 = Substitute. RQ = 6 Multiply each side by 9 and simplify.

EXAMPLE 2 Solve a real-world problem Shoerack On the shoerack shown, AB = 33 cm, BC = 27 cm, CD = 44 cm,and DE = 25 cm, Explain why the gray shelf is not parallel to the floor. SOLUTION Find and simplify the ratios of lengths determined by the shoerack. CD DE 44 25 = CB BA 27 33 = 9 11 Because , BD is not parallel to AE . So, the shelf is not parallel to the floor. 44 25 9 11 =

GUIDED PRACTICE for Examples 1 and 2 1. Find the length of YZ . 315 11 ANSWER 2. Determine whether PS || QR . ANSWER parallel

EXAMPLE 3 Use Theorem 6.6 City Travel In the diagram, 1, 2, and 3 are all congruent and GF = 120 yards, DE = 150 yards, and CD = 300 yards. Find the distance HF between Main Street and South Main Street. FG GH ED DC = SOLUTION Corresponding angles are congruent, so FE, GD , and HC are parallel. Use Theorem 6.6.

The distance between Main Street and South Main Street is 360 yards. EXAMPLE 3 Use Theorem 6.6 HG GF CD DE = Parallel lines divide transversals proportionally. HG + GF GF CD + DE DE = Property of proportions. HF 120 300 + 150 150 = Substitute. 450 150 HF 120 = Simplify. FG GH ED DC = Multiply each side by 120 and simplify. HF = 360 The distance between Main Street and South Main Street is 360 yards.

EXAMPLE 4 Use Theorem 6.7 In the diagram, QPR RPS. Use the given side lengths to find the length of RS . ~ SOLUTION Because PR is an angle bisector of QPS, you can apply Theorem 6.7. Let RS = x. Then RQ = 15 – x. RQ RS PQ PS = Angle bisector divides opposite side proportionally. 15 – x x = 7 13 Substitute. 7x = 195 – 13x Cross Products Property x = 9.75 Solve for x.

GUIDED PRACTICE for Examples 3 and 4 Find the length of AB. 3. 4. ANSWER 19.2 ANSWER 2 4