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Proportions and Similar Triangles 8.6

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Theorem 8.4 Triangle Proportionality Theorem If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally.

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Converse of the Triangle Proportionality Theorem If a line divides two sides of a triangle proportionally, then it is parallel to the third side.

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Example 1 In the diagram ll, BD=8,DC=4, and AE=12. What is the length of ? 4 812

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Example 2 Given the diagram, determine whether ll

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Find x. 36x = x = 23

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Find x. 2(x + 6) = 5x 2x + 12 = 5x -2x 12 = 3x 3 4 = x

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Find x. 2.5(8 – x) = 3.5x 20 – 2.5x = 3.5x +2.5x +2.5x 20 = 6x 6 3 = x

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Find x. 3x = x = x 3 4.5

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Find x. 6 x 10 x + 5 6(x + 5) = 10x 6x + 30 = 10x -6x 30 = 4x = x

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Find x. 12.8x = x = 9

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Find x. 24x = x = 10 in.

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Find x. 12x = x = 6 ft.

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Theorem 8.6 If three parallel lines intersect two transversals, then they divide the transversals proportionally.

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Example

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Theorem 8.7 If a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the two sides.

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Example x 14-x

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