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Understanding Proportional Parts of Triangles

Learn how to find lengths of segments and verify parallel lines using triangle proportionality theorems. Explore examples and corollaries related to parallel lines and proportional parts in triangles. Quiz included.

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Understanding Proportional Parts of Triangles

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  1. Parallel Lines andProportional Parts Section 6-4

  2. Proportional Parts of Triangles: • Non-Parallel transversals that intersect 2 Parallel lines can be extended to form 2 similar triangles. Line a║line b Line a Line b

  3. Example: Finding the Length of a Segment Find US. Since segment ST║segment UV, then ∆RST ~ ∆RUV.

  4. Example: Find PN. PN = 7.5

  5. Verify that . Since , by the Converse of the Triangle Proportionality Theorem. Example: Verifying Segments are Parallel

  6. AC = 36 cm, and BC = 27 cm. Verify that . Since , by the Converse of the Triangle Proportionality Theorem. Example:

  7. Midsegment in a Triangle: • Segment whose endpoints are the midpoints of 2 sides of a triangle. Triangle Midsegment Theorem: A midsegment of a triangle is║to one side and its length is half that side. 4 cm Parallel 8 cm

  8. Triangle Midsegment Theorem Corollaries: • If three or more║lines intersect two transversals, then they cut off the transversals proportionally. • If three or more║lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal.

  9. If lines AD, BE, and CF are ║, then: • AB/BC = DE/EF • AC/DF = BC/EF • AC/BC = DF/EF

  10. If lines AD, BE, and CF are ║ and AB  BC, then DE  EF

  11. Lesson Quiz: Part I Find the length of segment:

  12. Verify that BE and CD are parallel. Since , by the Converse of the ∆Proportionality Thm. Lesson Quiz: Part II

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