Geometry 8.4 Proportionality Theorems 8.4 Warmup Explain why the triangles are similar. Then find the value of x. 1. 2. 3. 4. 4 18 x 15 6 20 x x Hint: Use Pyth. Thm. January 18, 2019 Geometry 8.4 Proportionality Theorems

Geometry 8.6 Proportions and Similar Triangles 8.2 Practice A January 18, 2019 Geometry 8.6 Proportions and Similar Triangles

Geometry 8.6 Proportions and Similar Triangles 8.2 Practice A January 18, 2019 Geometry 8.6 Proportions and Similar Triangles

Geometry 8.6 Proportions and Similar Triangles 8.3 Practice A In Exercises 5 and 6, show that the triangles are similar and write a similarity statement. Explain your reasoning. January 18, 2019 Geometry 8.6 Proportions and Similar Triangles

8.4 Proportionality Theorems Geometry 8.4 Proportionality Theorems

Geometry 8.4 Proportionality Theorems 8.4 Essential Question What proportionality relationships exist in a triangle intersected by an angle bisector or by a line parallel to one sides? January 18, 2019 Geometry 8.4 Proportionality Theorems

Geometry 8.4 Proportionality Theorems Goals Use proportionality theorems to calculate segment lengths. Solve problems using these theorems. January 18, 2019 Geometry 8.4 Proportionality Theorems

Theorem 8.6 Triangle Proportionality Theorem If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally. 𝐼𝑓 𝐷𝐸 // 𝐴𝐵 , 𝑡ℎ𝑒𝑛 𝐴𝐷 𝐵𝐸 = 𝐷𝐶 𝐸𝐶 𝑜𝑟 𝐴𝐷 𝐷𝐶 = 𝐵𝐸 𝐸𝐶 A B C D E January 18, 2019 Geometry 8.4 Proportionality Theorems

Geometry 8.4 Proportionality Theorems Proof CDE ~ CAB. Why? AA~ C t s D E v r A B January 18, 2019 Geometry 8.4 Proportionality Theorems

Geometry 8.4 Proportionality Theorems Proof CDE ~ CAB. Recall the Addition Property of Proportions: C t s t + v D E r + s v r A B January 18, 2019 Geometry 8.4 Proportionality Theorems

Theorem 8.6 Simplified 𝐼𝑓 𝐷𝐸 // 𝐴𝐵 , 𝑡ℎ𝑒𝑛 𝑎 𝑏 = 𝑐 𝑑 𝑜𝑟 𝑎 𝑐 = 𝑏 𝑑 b a c 𝐼𝑓 𝐷𝐸 // 𝐴𝐵 , 𝑡ℎ𝑒𝑛 𝑎 𝑏 = 𝑐 𝑑 𝑜𝑟 𝑎 𝑐 = 𝑏 𝑑 January 18, 2019 Geometry 8.4 Proportionality Theorems

Geometry 8.4 Proportionality Theorems Example 1 Find a. 8 a 3 9 January 18, 2019 Geometry 8.4 Proportionality Theorems

Geometry 8.4 Proportionality Theorems Example 2 You do it. Find x. 7 5 x 12 January 18, 2019 Geometry 8.4 Proportionality Theorems

Geometry 8.4 Proportionality Theorems We did this before… Do you remember where? January 18, 2019 Geometry 8.4 Proportionality Theorems

Triangle Midsegment Theorem A midsegment of a triangle is parallel to the third side and ½ its length. R S But this was a special case where the parallel line divides the two sides in half. January 18, 2019 Geometry 8.4 Proportionality Theorems

Geometry 8.4 Proportionality Theorems Example 3 Find x. x 14 – x 6 10 January 18, 2019 Geometry 8.4 Proportionality Theorems

Geometry 8.4 Proportionality Theorems x 14 – x 6 10 16 14 January 18, 2019 Geometry 8.4 Proportionality Theorems

Theorem 8.7 Conv. of Δ Propor. Thm. If a line divides two sides of a triangle proportionally, then it is parallel to the third side. 𝐼𝑓 𝐴𝐷 𝐵𝐸 = 𝐷𝐶 𝐸𝐶 ,𝑡ℎ𝑒𝑛 𝐷𝐸 // 𝐴𝐵 A B C D E January 18, 2019 Geometry 8.4 Proportionality Theorems

Geometry 8.4 Proportionality Theorems Your Turn Problems Identify what theorem(s) apply. Write the proportion and solve. January 18, 2019 Geometry 8.4 Proportionality Theorems

Geometry 8.4 Proportionality Theorems Example 4 Is PQ || RS ? YES 8 6 P Q 9 12 R S January 18, 2019 Geometry 8.4 Proportionality Theorems

Geometry 8.4 Proportionality Theorems Your Turn 4 Is EB || DC? A B C D E 10 12 4 4.8 Yes: Sides are divided proportionally. January 18, 2019 Geometry 8.4 Proportionality Theorems

Geometry 8.4 Proportionality Theorems Example 5 Is AB || CD ? YES 6 9 24 16 A B 10 15 C D January 18, 2019 Geometry 8.4 Proportionality Theorems

Geometry 8.4 Proportionality Theorems Your Turn 5 Solve for x. A B C D E 10 15 x 8 January 18, 2019 Geometry 8.4 Proportionality Theorems

Theorem 8.8 Three Parallel Lines Thm If three (or more) parallel lines intersect two transversals, they divide the transversals proportionally. a c d b January 18, 2019 Geometry 8.4 Proportionality Theorems

Geometry 8.4 Proportionality Theorems Example 6 A B C D E M N O P January 18, 2019 Geometry 8.4 Proportionality Theorems

Geometry 8.4 Proportionality Theorems Example 7 Solve for x. 10 14 x x + 2 Set up the proportion and solve. January 18, 2019 Geometry 8.4 Proportionality Theorems

Geometry 8.4 Proportionality Theorems Example 7 Check it. 10 14 x x + 2 5 7 January 18, 2019 Geometry 8.4 Proportionality Theorems

Example 8 You do it. Solve for x. 24 20 January 18, 2019 Geometry 8.4 Proportionality Theorems

Geometry 8.4 Proportionality Theorems Example 9 In the diagram, ∠1, ∠2, and ∠3 are all congruent, GF = 120 yards, DE = 150 yards, and CD = 300 yards. Find HF, the distance between Main Street and South Main Street. January 18, 2019 Geometry 8.4 Proportionality Theorems

Theorem 8.9 Triangle Angle Bisector Thm. Theorem Demo If a ray bisects an angle of a triangle, then it divides the opposite side into segments that are proportional to the other sides. a b c d January 18, 2019 Geometry 8.4 Proportionality Theorems

Geometry 8.4 Proportionality Theorems Example 10 Solve for x. 6 8 x 4 January 18, 2019 Geometry 8.4 Proportionality Theorems

Geometry 8.4 Proportionality Theorems Example 11 Solve for x. 20 14 8 ? x What is this value in terms of x? January 18, 2019 Geometry 8.4 Proportionality Theorems

Geometry 8.4 Proportionality Theorems Example 11 Solve for x. 20 14 8 20 – x x Now, write the proportion and solve. January 18, 2019 Geometry 8.4 Proportionality Theorems

Example 12 You do it. Solve for a. 44 12 a 10 24 January 18, 2019 Geometry 8.4 Proportionality Theorems

Geometry 8.4 Proportionality Theorems Your Turn. Solve for a. a 8 4 6 January 18, 2019 Geometry 8.4 Proportionality Theorems

Geometry 8.4 Proportionality Theorems Your Turn. Solve for x. x 9 4 January 18, 2019 Geometry 8.4 Proportionality Theorems

Geometry 8.4 Proportionality Theorems Summary If a line parallel to a side of a triangle intersects the other two sides, it divides them proportionally. If a ray bisects an angle of a triangle, it divides the opposite side in proportion to the other two sides. January 18, 2019 Geometry 8.4 Proportionality Theorems

Geometry 8.4 Proportionality Theorems Homework January 18, 2019 Geometry 8.4 Proportionality Theorems