Exercise 17 Page 578

We are given a triangle, △ZWX, and we want to determine whether segments VY and ZW are parallel.

By the Converse Triangle Proportionality Theorem, if a line intersects two sides of a triangle and separates the sides into proportional corresponding segments, then the line is parallel to the third side of the triangle. Let's check if the sides are divided proportionally. ZV/VX ? = WY/YX We have to find and simplify each ratio. To find the left-hand side, let ZV=x. Since ZX=4x and ZV+VX=ZX, VX=3x. To find the right-hand side, we are given lengths of YX and WX. Let's find the length of WY. Now we can compare both sides of our proportion. ZV/VX ? = WY/YX ⇕ 1/3 ≠ 10/21 As we can see, the sides are not divided proportionally. Therefore, segments VY and ZW are not parallel.