Parallel Lines and Proportional Parts By: Jacob Begay

Theorem 7-4 Triangle Proportionality: If a line is parallel 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. If a line is parallel 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. A B C D E B C D A C E CB CA CD CE = BD AE =

Theorem 7-5 Converse of the Triangle Proportionality: If a line intersects two sides of a triangle and separates the sides into corresponding segments of proportional lengths, then the line is parallel to the third side. If a line intersects two sides of a triangle and separates the sides into corresponding segments of proportional lengths, then the line is parallel to the third side. A B C D E BDAE

Theorem 7-6 Triangle Midpoint Proportionality: A segment whose endpoints are the midpoints of two sides of a triangle is parallel to the third side of the triangle, and its length is one-half the length of the third side. A segment whose endpoints are the midpoints of two sides of a triangle is parallel to the third side of the triangle, and its length is one-half the length of the third side. A B C D E 2BD=AE OR BD=1/2AE BD ll AE

Corollary 7-1 If three or more parallel lines intersect two transversals, then they cut off the transversals proportionally. If three or more parallel lines intersect two transversals, then they cut off the transversals proportionally. A B C D E F G BC EF CD FG AB AE AD AG = = AC AF BC EF CD AE FG AB = =

Corollary 7-2 If three or more parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal. If three or more parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal. B C D E F G BECFGD

Example Based on the figure below, which statement is false? Based on the figure below, which statement is false? A D B C E A.DE is Parallel to BCC.ABC ~ ADE B.D is the Midpoint of ABD.ABC is congruent to ADE D. ABC is congruent to ADE. Corresponding sides of the triangles are proportional but not congruent.

Example Find the value of X so that PQ is parallel to BC. Find the value of X so that PQ is parallel to BC. A P Q BC X+0.25 A.1C.1.25 B.2.5 D.2 D. 2 Since the corresponding segments must be proportional for PQ to be parallel to BC.

Example Triangle ABC has vertices A (0,2), B (12,0), and C (2,10). Triangle ABC has vertices A (0,2), B (12,0), and C (2,10). A. Find the coordinates of D, the midpoint of Segment AB, and E, the midpoint of Segment CB. A. Find the coordinates of D, the midpoint of Segment AB, and E, the midpoint of Segment CB. B. Show that DE ll AC. B. Show that DE ll AC. C. Show that 2DE = AC. C. Show that 2DE = AC. D 0+12, 2+02 =Or D = (6,1) E 12+2, = Or E = (7,5) Midpoint Segment AB (6,1) Midpoint Segment CB (7,5) Slope of AC = AC=4 Slope of DE = DE=4 AC ll DE AC=(0-2) + (2-10) 22 =4+64 = 68 Or 217 DE= (6-7) + (1-5) 22 = 1+16 Or 17 Therefore 2DE = AC