Geometry 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

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

Theorem 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

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

Triangle Mid-segment Theorem Midsegment: A segment with endpoints that are midpoints of two sides of the triangle. A midsegment 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. 20 a b c x 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. 495-496 1-6, 8-21, 24-27, 35-38