1. Lesson 5-4: Proportional Parts 1 Proportional Parts Lesson 5-4

2. Lesson 5-4: Proportional Parts 2 Two polygons are similar if and only if their corresponding angles are congruent and the measures of their corresponding sides are proportional. AB C DE F Similar Polygons

3. Lesson 5-4: Proportional Parts 3 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 length. Triangle Proportionality Theorem 12 34A B C D E Converse: 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.

4. Lesson 5-4: Proportional Parts 4 4x + 3 9 A B C D E 2x + 3 5 A BC D E If BE = 6, EA = 4, and BD = 9, find DC. 6x = 36 x = 6 Solve for x. Example 1: Example 2: Examples……… 6 4 9 x

5. Lesson 5-4: Proportional Parts 5 Theorem A segment that joins the midpoints of two sides of a triangle is parallel to the third side of the triangle, and its length is one-half the length of the third side. R S T M L

6. Lesson 5-4: Proportional Parts 6 (1) then the perimeters are proportional to the measures of the corresponding sides. If two triangles are similar:

7. Lesson 5-4: Proportional Parts 7 A B C D E F The perimeter of ΔABC is 15 + 20 + 25 = 60. Side DF corresponds to side AC, so we can set up a proportion as: Given: ΔABC ~ ΔDEF, AB = 15, AC = 20, BC = 25, and DF = 4. Find the perimeter of ΔDEF. Example: 15 20 25 4