Exercise 15 Page 578

We are given a triangle, △ZWX, and we want to determine whether the 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/YXSince we are given the lengths of VX and ZX, we can find the length of ZV.

ZV+VX=ZX

SubstituteII

VX= 7.5, ZX= 24

ZV+ 7.5= 24

SubEqn

LHS-7.5=RHS-7.5

ZV=16.5

Similarly, we have the lengths of WY and WX. Let's find the length of YX.

WY+YX=WX

SubstituteII

WY= 27.5, WX= 40

27.5+YX= 40

SubEqn

LHS-27.5=RHS-27.5

YX=12.5

Now we can substitute all known lengths into our proportion. ZV/VX ? = WY/YX ⇕ 16.5/7.5 ? = 27.5/12.5 We will simplify each ratio as much as possible to see if they are equivalent.

16.5/7.5 ? = 27.5/12.5

ReduceFrac

a/b=.a /1.5./.b /1.5.

11/5 ? = 27.5/12.5

ReduceFrac

a/b=.a /2.5./.b /2.5.

11/5 = 11/5 ✓

As we can see, the sides are divided proportionally. Therefore, segments VY and ZW are parallel.