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Working with Ratio Segments: Triangle Proportionality Theorem

Learn how to use the Triangle Proportionality Theorem to find and determine the length of segments in similar triangles. Understand the concepts of parallel lines, segment congruence, and angle bisectors.

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Working with Ratio Segments: Triangle Proportionality Theorem

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  1. Introduction Archaeologists, among others, rely on the Angle-Angle (AA), Side-Angle-Side (SAS), and Side-Side-Side (SSS) similarity statements to determine actual distances and locations created by similar triangles. Many engineers, surveyors, and designers use these statements along with other properties of similar triangles in their daily work. Having the ability to determine if two triangles are similar allows us to solve many problems where it is necessary to find segment lengths of triangles. 5.4.2: Working with Ratio Segments

  2. Key Concepts Parallel lines are lines in a plane that either do not share any points and never intersect, or share all points. If a line parallel to one side of a triangle intersects the other two sides of the triangle, then the parallel line divides these two sides proportionally. This is known as the Triangle Proportionality Theorem. 5.4.2: Working with Ratio Segments

  3. Key Concepts, continued In the figure above, ; therefore, . 5.4.2: Working with Ratio Segments

  4. Key Concepts, continued Notice the arrows in the middle of and , which indicate the segments are parallel. This theorem can be used to find the lengths of various sides or portions of sides of a triangle. because of the Side-Angle-Side (SAS) Similarity Statement. It is also true that if a line divides two sides of a triangle proportionally, then the line is parallel to the third side, as shown on the next slide. 5.4.2: Working with Ratio Segments

  5. Key Concepts, continued In the figure above, ; therefore, . 5.4.2: Working with Ratio Segments

  6. Key Concepts, continued This is helpful when determining if two lines or segments are parallel. It is possible to determine the lengths of the sides of triangles because of the Segment Addition Postulate. This postulate states that if B is between A and C, then AB + BC = AC. 5.4.2: Working with Ratio Segments

  7. Key Concepts, continued It is also true that if AB + BC = AC, then B is between A and C. 5.4.2: Working with Ratio Segments

  8. Key Concepts, continued Segment congruence is also helpful when determining the lengths of sides of triangles. The Reflexive Property of Congruent Segments means that a segment is congruent to itself, so According to the Symmetric Property of Congruent Segments, if , then . The Transitive Property of Congruent Segments allows that if and , then . 5.4.2: Working with Ratio Segments

  9. Key Concepts, continued This information is also helpful when determining segment lengths and proving statements. If one angle of a triangle is bisected, or cut in half, then the angle bisector of the triangle divides the opposite side of the triangle into two segments that are proportional to the other two sides of the triangle. This is known as the Triangle Angle Bisector Theorem. 5.4.2: Working with Ratio Segments

  10. Key Concepts, continued 5.4.2: Working with Ratio Segments

  11. Key Concepts, continued In the figure on the previous slide, ; therefore, . These theorems can be used to determine segment lengths as well as verify that lines or segments are parallel. 5.4.2: Working with Ratio Segments

  12. Common Errors/Misconceptions assuming a line parallel to one side of a triangle bisects the remaining sides rather than creating proportional sides interchanging similarity statements with congruence statements 5.4.2: Working with Ratio Segments

  13. Guided Practice Example 1 Find the length of . 5.4.2: Working with Ratio Segments

  14. Guided Practice: Example 1, continued Identify the given information. According to the diagram, . 5.4.2: Working with Ratio Segments

  15. Guided Practice: Example 1, continued Find the length of . Use the Triangle Proportionality Theorem to find the length of . 5.4.2: Working with Ratio Segments

  16. Guided Practice: Example 1, continued The length of is 8.25 units. ✔ 5.4.2: Working with Ratio Segments

  17. Guided Practice: Example 1, continued 5.4.2: Working with Ratio Segments

  18. Guided Practice Example 4 Is ? 5.4.2: Working with Ratio Segments

  19. Guided Practice: Example 4, continued Determine if divides and proportionally. 5.4.2: Working with Ratio Segments

  20. Guided Practice: Example 4, continued State your conclusion. ; therefore, is not parallel to because of the Triangle Proportionality Theorem. ✔ 5.4.2: Working with Ratio Segments

  21. Guided Practice: Example 4, continued 5.4.2: Working with Ratio Segments

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