Chapter 3.2, Problem 10WE

To determine

To find: the values of x and y .

Answer to Problem 10WE

  $x=70$

  $y=60$

Explanation of Solution

Given:

The diagram:

Concept Used:

Postulate 10: If two parallel lines are cut by a transversal then the corresponding angles are congruent

Theorem 2-3: Vertically opposite angles are congruent.

Label the given figure as shown below:

Now, lines $\overline{AB}\parallel \overline{CD}$ and $\overline{GC}$ is a transversal, so by postulate 10,

  $\angle GEA\cong \angle ECD$ (Corresponding angles)

Thus,

  $\begin{array}{c}m\angle GEA=m\angle ECD\\ x=70\end{array}$

Also, lines $\angle AED$ and $\angle FEB$ are vertically opposite angles, so by theorem 2-3,

  $\angle AED\cong \angle FEB$ (Corresponding angles)

Thus,

  $m\angle AED=m\angle FEB=50$

From the above figure, it can be seen that the sum of the angles $\angle GEA$ , $\angle AED$ and $\angle DEC$ forms the straight line $\overline{GC}$ , so these angles are supplementary angles. That is,

  $\begin{array}{c}m\angle GEA+m\angle AED+m\angle DEC=180\\ 70+50+y=180\\ 120+y=180\end{array}$

  $\begin{array}{c}y=180-120\\ y=60\end{array}$