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Geometry Theorems and Proportions: Explained Examples and Homework Help

Learn about the Triangle Proportionality Theorem, Parallel Lines, and more with detailed examples and exercises for practice. Solve problems involving parallels, midsegments, and transversals. Improve your understanding of geometry concepts.

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Geometry Theorems and Proportions: Explained Examples and Homework Help

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  1. C B D E A Geometry7.4 Parallel Lines and Proportional Parts • Triangle Proportionality Theorem (Theorem 7.4) • If a line is || 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.

  2. In the figure, AE || BD. Find the value of x. E 8 D x + 5 C x B 6 A Example

  3. C B D E A Theorem 7.5 • If a line intersects two sides of a triangle and separates the sides into corresponding segments of proportional lengths, then the line is || to the third side. then BD || AE

  4. A 6 8 D E 4 3 B C Example • Determine whether DE || BC. • Yes because 6/3 = 8/4

  5. A E D B C Theorem 7.6: Triangle Midsegment Theorem • Midsegment: A segment with endpoints that are midpoints of two sides of the triangle. • A midsegment of a triangle is || to one side of the triangle and its length is one-half the length of the third side.

  6. Example • Refer to the figure and Example #3 on page 407 • The example uses the midpoint formula, the slope formula and the distance formula to verify coordinates of midpoint, parallelism, and lengths of segments.

  7. X A D B E C F Corollary 7.1 • If 3 or more || lines intersect 2 transversals, then they cut off the transversals proportionally.

  8. 15 9 12 x a b c Example • In the figure, a || b || c. Find the value of x. • 20

  9. Corollary 7.2 • If 3 or more || lines cut off  segments on one transversal, then they cut off  segments on every transversal.

  10. Homework #48 • p. 411 13-18, 21-29 odd, 32-38 even, 54-55

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