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

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Example 2 Given the diagram, determine whether ll . 21 56 16 48

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Find x. 36x = 840 x = 23⅓

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

3⅓ = x

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Find x. 7 x 3x = 31.5 3 4.5 x = 10.5

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**Find x. 6(x + 5) = 10x 6x + 30 = 10x -6x -6x 30 = 4x 4 4 7.5 = x x 6**

7.5 = x

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Find x. 12.8x = 115.2 x = 9

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Find x. 24x = 240 x = 10 in.

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Find x. 12x = 72 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 14-x x

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6.6 – Use Proportionality Theorems. Triangle Proportionality Theorem If a line parallel to one side of a triangle intersects the other two sides, then.

6.6 – Use Proportionality Theorems. Triangle Proportionality Theorem If a line parallel to one side of a triangle intersects the other two sides, then.

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