Lesson 7 – 4 Parallel Lines and Proportional Parts Geometry Lesson 7 – 4 Parallel Lines and Proportional Parts Objective: Use proportional parts within triangles. Use proportional parts with parallel lines.

Theorem Triangle Proportionality Theorem If a line is parallel to one side of a triangle and intersects the other two sides, then it divides the sides into segments of proportional lengths.

In triangle PQR ST II RQ. If PT = 7. 5, TQ = 3, and SR = 2 In triangle PQR ST II RQ. If PT = 7.5, TQ = 3, and SR = 2.5, find PS and PR. Method from last section: 7.5 x 3 3x = 18.75 2.5 x = 6.25 10.5x = 7.5x + 18.75 PS = 6.25 3x = 18.75 PR = 8.75 x = 6.25

If PS = 12.5, SR = 5, and PT = 15, find TQ. x 5 12.5x = 75 x = 6 TQ = 6

Theorem Converse of Triangle Proportionality Theorem If a line intersects two sides of a triangle and separates the sides into proportional corresponding segments, then the line is parallel to the third side of the triangle.

In triangle DEF, EH = 3, HF = 9, and DG is one-third the length of GF In triangle DEF, EH = 3, HF = 9, and DG is one-third the length of GF. Is DE II GH? 3 x 9 3x 9x = 9x The sides are proportional, therefore DE II GH

DG is half the length of GF, EH = 6, and HF = 10. IS DE II GH? x 10 2x 10x = 12x The lines are not parallel.

Midsegment Midsegment of a triangle A segment with endpoints that are the midpoints of two sides of the triangle.

Theorem Triangle Midsegment Theorem A midsegment of a triangle is parallel to one side of the triangle, and its length is one half the length of that side.

XY and XZ are midsegments of the triangle. Find XZ ST 6.5 7 6.5 7 14 124o 6.5 6.5 124

Find each measure DE DB 7.5 9.2 7.5 7.5 7.5 9.2 82o 9.2 82

Proportional Parts of Parallel Lines If three or more parallel lines intersect two transversals, then they cut off the transversals proportionally.

Frontage is the measurement of a property’s boundary that runs along the side of a particular feature such as a street, lake, ocean, or river. Find the ocean frontage for Lot A to the nearest tenth of a yard. 42x = 3480 x 82.9 Lot A is approximately 82.9 yards

Corollary Congruent Parts of Parallel Lines If three or more parallel lines cut off congruent segments on one transversal, then they cute off congruent segments on every transversal.

Find x and y 6x – 5 = 4x + 3 3y + 8 = 5y - 7 2x = 8 15 = 2y 7.5 = y

Find x 2x + 1 = 3x - 5 6 = x 3x = 5 x = 5/3

Homework Pg. 489 1 – 9 all, 10 – 26 E, 54 – 66 E