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Parallel Lines and Proportional Parts

Parallel Lines and Proportional Parts. By: Jacob Begay. CD. BD. CE. AE. CB. CA. Theorem 7-4 Triangle Proportionality:.

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Parallel Lines and Proportional Parts

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  1. Parallel Lines and Proportional Parts By: Jacob Begay

  2. CD BD CE AE CB CA 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. C C C D B B D A A E E = =

  3. 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. C BD AE D B E A

  4. C D B A E 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. BD ll AE 2BD=AE OR BD=1/2AE

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

  6. 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. D C BE CF GD B E F G

  7. A 4 3 E D 4 3 C B Example • Based on the figure below, which statement is false? A.DE is Parallel to BC C.ABC ~ ADE B.D is the Midpoint of AB D.ABC is congruent to ADE D. ABC is congruent to ADE. Corresponding sides of the triangles are proportional but not congruent.

  8. Example • Find the value of X so that PQ is parallel to BC. A 4 3 Q P X+0.25 3 B C A.1 C.1.25 B.2.5 D.2 D. 2 Since the corresponding segments must be proportional for PQ to be parallel to BC.

  9. 0+12, 2+0 D = Or D = (6,1) 2 2 12+2, 0+10 E Or E = (7,5) = 2 2 Slope of AC = 2-10 Slope of DE = 1-5 0-2 6-7 DE = 2 2 (6-7) + (1-5) 2 2 AC= (0-2) + (2-10) = = 4+64 1+16 Or 17 = Or 68 2 17 Example • 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. • B. Show that DE ll AC. • C. Show that 2DE = AC. Midpoint Segment AB (6,1) Midpoint Segment CB (7,5) AC ll DE AC=4 DE=4 Therefore 2DE = AC

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