4.3: Theorems about Proportionality WELCOME 4.3: Theorems about Proportionality Last Night’s HW: 4.2 Handout Tonight’s HW: 4.3 Handout

Warm Up 1. Are the triangles similar? If so explain how you know. Write a similarity statement: ∆______~∆_______ 2. Given the figures are similar. Find the missing side length.

Homework Review

Proof Review

Chapter 4.3 Learning Targets

Angle Bisector Proportionality If a ray bisects an angle of a triangle, then it divides the side into segments whose lengths are proportional to the lengths of the other two sides. A D C B AD CA If CD bisects ∠ ACB, then = DB CB

Example 2: Example 1:

Triangle Proportionality If a line is ‖ to a side of a ∆ & intersects the other sides, then it divides them proportionally. If a line divides two sides of a ∆ proportionally, then it is parallel to the third side. Q Q T T R R U U S S RT RU RT RU If TU ‖ QS, then = If = , then TU ‖ QS TQ US TQ US

Example 3: Example 4:

Parallel Line Proportionality If three parallel lines intersect two transversals, then they divide the transversals proportionally. s t r U l W Y Z m X V UW VX If r ‖ s and s ‖ t, then = WY XZ

Example 5:

Midsegment of a Triangle A segment that connects the midpoints of two sides of a ∆ D F E DE , EF , FD are Midsegment

Theorem 5.9 Midsegment Theorem The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long. C E D 8 A B 16 1 2 DE ‖ AB and DE = AB

Example 6: Find x