Triangle Proportionality Theorems & Midsegment Properties

1. PARALLEL LINES AND PROPORTIONAL PARTS • Use proportional parts of triangles • Divide a segment into parts

5. C B D A E

6. TRIANGLE PROPORTIONALITY THEOREM If a line is parallel to one side of a triangle and intersects the other two sides at two distinct points, then it separates these sides into segments of proportional lengths. C Example: B D A E

7. Example 1– Find the Length of a Side Find x From the Triangle Proportionality Theorem, E F 6 9 Substitute the known measures: H L x 21 Cross products Multiply Divide each side by 9 G

8. CONVERSE OF THE TRIANGLE PROPORTIONALITY THEOREM 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. C Example: D B E A

9. DEFINITION A Midsegment of a triangle is a segment whose endpoints are the midpoints of two sides of the triangle. C A Midsegment D B E A

10. TRIANGLE MIDSEGMENT THEOREM A midsegment of a triangle is parallel to one side of the triangle and its length is one half the length of that side. C D B E A

11. Example 2– Find x and y

12. Example 3– Find x