# Parallel Lines and Proportional Parts

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Parallel Lines and Proportional Parts
Lesson 6-4 Parallel Lines and Proportional Parts

Ohio Content Standards:

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

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

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.

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

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

Theorem 6.4 Triangle Proportionality Theorem

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.

Theorem 6.4 Triangle Proportionality Theorem
C B D A E

R 8 V 3 S x U 12 T

Theorem 6.5 Converse of the Triangle Proportionality Theorem

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.

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

D G H F E

Midsegment

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

Theorem 6.6 Triangle Midsegment Theorem

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.

Theorem 6.6 Triangle Midsegment Theorem
C D B E A

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

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

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

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

Corollary 6.1

Corollary 6.1 If three or more parallel lines intersect two transversals, then they cut off the transversals proportionally.

Corollary 6.1 D F E C B A

Corollary 6.2

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.

Corollary 6.2 D F E C B A

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

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

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