Lesson 7-6 Proportional Lengths (page 254) Essential Question How do you calculate the lengths of sides of similar triangles?

If AB : BC = XY : YZ, then AC and XZ are said to be divided proportionally . A B C X Y Z

Theorem 7-3 Triangle Proportionality Theorem ➤ ➤ If a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally . T Given: ∆ RST; PQ || RS Prove: P ➤ Q ➤ S R

Triangle Proportionality Theorem Given: ∆ RST; PQ || RS Prove: PT QT RP SQ T Top Left Top Right P ➤ Q Bottom Left Bottom Right ➤ S R

many equivalent proportions can be justified … NOTE: Since ∆RST ~ ∆PQT, many equivalent proportions can be justified … T P ➤ Q ➤ S R by the AA ~ Postulate

➤ ➤ Here are many equivalent proportions along with informal statements describing them. P ➤ Q ➤ S R

T P ➤ Q ➤ S R

T P ➤ Q ➤ S R

T P ➤ Q ➤ S R

This is the only way to get the parallel sides. ➤ Q ➤ S R DO NOT FORGET THESE RATIOS! This is the only way to get the parallel sides.

Example #1. Find the value of “x”. 20 15 ➤ 8 x ➤

Theorem 5-9 - REVIEW!!! If three parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal. Given: Prove: A ➤ X B ➤ Y C ➤ Z

Corollary If three parallel lines intersect two transversals, then they divide the transversals proportionally . R ➤ X Given: Prove: S ➤ Y T ➤ Z

➤ ➤ ➤ Example #2. Find the value of “x”. 25 - x = 20 25 - x 16 x = 5 x 4 ➤

Theorem 7-4 Triangle Angle-Bisector Theorem DG bisects ∠FDE If a ray bisects an angle of a triangle, then it divides the opposite side into segments proportional to the other two sides . F Given: ∆ DEF ; DG bisects ∠FDE Prove: G E D

Example #3. Find the value of “x”. 27 45 x = 15 40 - x = 25 40

Example #4. Find the value of “x”. 12 18 x 24

How do you calculate the lengths of sides of similar triangles? Assignment Written Exercises on pages 272 & 273 DO NOW: 1 to 11 ALL numbers GRADED: 13, 15, 17, 21, 23 YOU MUST SHOW YOUR WORK & DIAGRAM! Prepare for Quiz on Lessons 7-4 to 7-6 How do you calculate the lengths of sides of similar triangles?

How do you use similar polygons to solve real life problems? Prepare for Test on Chapter 7: Similar Polygons Chapter Review on pages 277 & 278: 4, 12, 16 – 20, 22 - 24 Chapter Test on page 279: 1, 10 - 14 How do you use similar polygons to solve real life problems?