# Geometry 7 4 Parallel Lines and Proportional Parts

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Geometry 7. 4 Parallel Lines and Proportional Parts • Triangle Proportionality Theorem (Theorem 7. 4) • 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. C B A D E

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

Theorem 7. 5 • 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. C then BD || AE A B D E

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

Theorem 7. 6: Triangle Midsegment Theorem • Midsegment: 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. A D B E C

Example • Refer to the figure and Example #3 on page 407 • The example uses the midpoint formula, the slope formula and the distance formula to verify coordinates of midpoint, parallelism, and lengths of segments.

Corollary 7. 1 • If 3 or more || lines intersect 2 transversals, then they cut off the transversals proportionally. X A B C D E F

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

Corollary 7. 2 • If 3 or more || lines cut off segments on one transversal, then they cut off segments on every transversal.

Homework #48 • p. 411 13 -18, 21 -29 odd, 32 -38 even, 54 -55