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**Parallel Lines and Proportional Parts**

Lesson 6-4 Parallel Lines and Proportional Parts

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**Ohio Content Standards:**

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**Ohio Content Standards:**

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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**Theorem 6.4 Triangle Proportionality Theorem**

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**

C B D A E

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R 8 V 3 S x U 12 T

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**Theorem 6.5 Converse of the Triangle Proportionality Theorem**

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**Theorem 6.5 Converse of the Triangle Proportionality Theorem**

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**

D B E A

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D G H F E

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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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**Theorem 6.6 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.

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**Theorem 6.6 Triangle Midsegment Theorem**

C D B E A

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**Triangle ABC has vertices A(-2, 2), B(2, 4), and C(4, -4)**

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

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**Triangle ABC has vertices A(-2, 2), B(2, 4), and C(4, -4)**

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

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**Triangle ABC has vertices A(-2, 2), B(2, 4), and C(4, -4)**

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

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**Triangle ABC has vertices A(-2, 2), B(2, 4), and C(4, -4)**

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

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Corollary 6.1

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Corollary 6.1 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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Corollary 6.2 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**

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. 26 13 16 Larch x Maple Hatch

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

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

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