Exercise 14 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/YXSince we are given the lengths of ZV and ZX, we can find the length of VX.

ZV+VX=ZX

SubstituteII

ZV= 6, ZX= 18

6+VX= 18

SubEqn

LHS-6=RHS-6

VX=12

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

WY+YX=WX

SubstituteII

YX= 16, WX= 24

WY+ 16= 24

SubEqn

LHS-16=RHS-16

WY=8

Now we can substitute all known lengths into our proportion. ZV/VX ? = WY/YX ⇕ 6/12 ? = 8/16 We will simplify each ratio as much as possible to see if they are equivalent.

6/12 ? = 8/16

ReduceFrac

a/b=.a /6./.b /6.

1/2 ? = 8/16

ReduceFrac

a/b=.a /8./.b /8.

1/2 = 1/2 ✓

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