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Lesson 6-4 Parallel Lines and Proportional Parts

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Ohio Content Standards:

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Estimate, compute and solve problems involving real numbers, including ratio, proportion and percent, and explain solutions.

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Ohio Content Standards: Estimate, compute and solve problems involving rational numbers, including ratio, proportion and percent, and judge the reasonableness of solutions.

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Ohio Content Standards: Use proportional reasoning and apply indirect measurement techniques, including right triangle trigonometry and properties of similar triangles, to solve problems involving measurements and rates.

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Ohio Content Standards: Use scale drawings and right triangle trigonometry to solve problems that include unknown distances and angle measures.

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Ohio Content Standards: Apply proportional reasoning to solve problems involving indirect measurements or rates.

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Theorem 6.4 Triangle Proportionality Theorem

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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.

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Theorem 6.4 Triangle Proportionality Theorem A D C B E

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T R S V U 8 12 x 3

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Theorem 6.5 Converse of the Triangle Proportionality Theorem

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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.

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Theorem 6.5 Converse of the Triangle Proportionality Theorem A D C B E

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D E H F G

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Midsegment

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Midsegment A segment whose endpoints are the midpoints of two sides of the triangle.

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Theorem 6.6 Triangle Midsegment Theorem

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

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Theorem 6.6 Triangle Midsegment Theorem A D C B E

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Triangle ABC has vertices A(-2, 2), B(2, 4), and C(4, -4). DE is a midsegment of ABC. B D E A C x y O

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Find the coordinates of D and E. B D E A C x y O

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Triangle ABC has vertices A(-2, 2), B(2, 4), and C(4, -4). DE is a midsegment of ABC. Verify that BC ll DE. B D E A C x y O

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Triangle ABC has vertices A(-2, 2), B(2, 4), and C(4, -4). DE is a midsegment of ABC. Verify that DE = ½ BC. B D E A C x y O

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Corollary 6.1

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If three or more parallel lines intersect two transversals, then they cut off the transversals proportionally.

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Corollary 6.1 D F E C B A

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Corollary 6.2

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If three or more parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal.

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Corollary 6.2 D F E C B A

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In the figure, Larch, Maple, and Hatch Streets are all parallel. The figure shows the distances in blocks that the streets are apart. Find x. x Maple 16 1326 Hatch Larch

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Find x and y. 5y5y 2x + 2 3x - 4

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Assignment: Pgs. 312-315 14-28 evens, 50-56 evens

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