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Geometry

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

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Geometry

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

  2. 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.

  3. Method from last section: 7.5 • In triangle PQR ST II RQ. If PT = 7.5, TQ = 3, and SR = 2.5, find PS and PR. 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

  4. 15 12.5 • If PS = 12.5, SR = 5, and PT = 15, find TQ. x 5 12.5x = 75 x = 6 TQ = 6

  5. 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.

  6. 3 • In triangle DEF, EH = 3, HF = 9, and DG is one-third the length of GF. Is DE II GH? x 9 3x 9x = 9x The sides are proportional, therefore DE II GH

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

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

  9. 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.

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

  11. Find each measure • DE • DB 7.5 9.2 7.5 9.2 7.5 7.5 82o 9.2 82

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

  13. 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

  14. 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.

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

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

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

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