Exercise 34 Page 599

We are given a triangle △ABE and we want to determine whether the segments AB and CD 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. AC/CE ? = BD/DEWe are given that AC=7 and AE=15. We can use this information to find the length of CE.

AC+CE=AE

SubstituteII

AC= 7, AE= 15

7+CE= 15

SubEqn

LHS-7=RHS-7

CE=8

Similarly, we are given that BD=10.5 and BE=22.5. Let's find length of DE.

BD+DE=BE

SubstituteII

BD= 10.5, BE= 22.5

10.5+DE= 22.5

SubEqn

LHS-10.5=RHS-10.5

DE=12

Now we can substitute all known lengths to our proportion. AC/CE ? = BD/DE ⇔ 7/8 ? = 10.5/12 We will simplify each ratio as much as possible to see if they are equivalent.

7/8 ? = 10.5/12

ReduceFrac

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

7/8 = 7/8 ✓

As we can see, the sides are divided proportionally. Therefore, the segments AB and CD are parallel.