1. Geometry 7.4 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. E A B C D

2. Example In the figure, AE || BD. Find the value of x. E 8 D x + 5 C x6
x + 5 x

3. 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. E A B C D
then BD || AE

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

5. 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. B D A C E

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

8. Example In the figure, a || b || c. Find the value of x. 20 a b c x 1215 9

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 , odd, even, 54-55