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Proportionality Theorems in Parallel Lines and Triangles

Explore and apply various theorems involving proportions with parallel lines and triangles. Discover the Triangle Proportionality Theorem, Converse of the Triangle Proportionality Theorem, Proportionality of Parallel Lines, and the Angle Bisector Proportionality Theorem.

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Proportionality Theorems in Parallel Lines and Triangles

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  1. Warm-Up 1 In the diagram, DE is parallel to AC. Name a pair of similar triangles and explain why they are similar. DBE ~ ABC AA (corresponding angles are congruent when parallel lines are cut by a transversal)

  2. Warm-Up 2 In the diagram, notice that AC divides the sides of the PBD proportionally. In other words, . What relationship exists between AC and BD? Are they parallel?

  3. Warm-Up 3 In the diagram, lines AD, BE, and CF are parallel. What relationship exists between AB, DE, BC, and EF?

  4. Warm-Up 4 Ray AD is an angle bisector. Notice that it divides the third side of the triangle into two parts. Are those parts congruent? Or is there some other relationship between them?

  5. 6.6: Use Proportionality Theorems Objectives: • To discover, present, and use various theorems involving proportions with parallel lines and triangles

  6. Proportionality Theorems! 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.

  7. Example 1 Find the length of YZ. 28.64

  8. Proportionality Theorems! Converse of the Triangle Proportionality Theorem If a line divides two sides of a triangle proportionally, then it is parallel to the third side.

  9. Example 3 Determine whether PS || QR. Yes

  10. Example 4 Find the value of x so that x=30

  11. Proportionality Theorems! If three parallel lines intersect two transversals, then they divide the transversals proportionally. Click hereto investigate these realtionships

  12. Example 5 Find the length of AB.

  13. Example 6 Find the value of x. 12.5

  14. Proportionality Theorems! Angle Bisector Proportionality Theorem If a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the other two sides.

  15. Example 7 Find the value of x. x=10

  16. Example 8 Find the value of x. 9.75

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