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Lesson 6-4 Parallel Lines and Proportional Parts.

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1 Lesson 6-4 Parallel Lines and Proportional Parts

2 Ohio Content Standards:

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

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

5 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.

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

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

8 Theorem 6.4 Triangle Proportionality Theorem

9 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.

10 Theorem 6.4 Triangle Proportionality Theorem A D C B E

11 T R S V U 8 12 x 3

12 Theorem 6.5 Converse of the Triangle Proportionality Theorem

13 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.

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

15 D E H F G

16 Midsegment

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

18 Theorem 6.6 Triangle Midsegment Theorem

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

20 Theorem 6.6 Triangle Midsegment Theorem A D C B E

21 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

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

23 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

24 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

25 Corollary 6.1

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

27 Corollary 6.1 D F E C B A

28 Corollary 6.2

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

30 Corollary 6.2 D F E C B A

31 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 Hatch Larch

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

33 Assignment: Pgs evens, evens


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