7.4 Parallel Lines and Proportional Parts Triangle Proportionality Theorem If a line is || to one side of a triangle and intersects the other two sides in two distinct points, then it separates these sides into segments of proportional lengths. E A B C D

In the figure, AE || BD. Find the value of x. Example In the figure, AE || BD. Find the value of x. B C D A E 8 6 x + 5 x

Theorem: Converse If a line intersects two sides of a triangle and separates the sides into corresponding segments of proportional lengths, then the line is || to the third side. E A B C D then BD || AE

Determine whether DE || BC. Example Determine whether DE || BC. Yes because 6/3 = 8/4 B D A C E 6 8 3 4

Triangle Mid-segment Theorem Mid-segment: A segment with endpoints that are midpoints of two sides of the triangle. A mid-segment of a triangle is || to one side of the triangle and its length is one-half the length of the third side. B D A C E

Corollary If 3 or more || lines intersect 2 transversals, then they cut off the transversals proportionally. A D X B C E F

Example In the figure, a || b || c. Find the value of x. X = 20 a b c 12 15 9

Corollary If 3 or more || lines cut off  segments on one transversal, then they cut off  segments on every transversal.

HOMEWORK Pgs. 494-496 #’S 1-6, 8-21, 24-27, 35-38