Chapter 3.2, Problem 9WE

To determine

To find: the values of x and y .

Answer to Problem 9WE

  $x=60$

  $y=18$

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 3-3: If two parallel lines are cut by a transversal then the same-side interior angles are supplementary.

Label the given figure as shown below:

From the above figure, it can be seen that $\angle BAH$ and $\angle BAE$ are supplementary angles. That is,

  $m\angle BAH+m\angle BAE=180$

Now substitute $m\angle BAH=120$ in the above equation,

  $\begin{array}{c}120+m\angle BAE=180\\ m\angle BAE=180-120\\ m\angle BAE=60\end{array}$

Now, lines $\overline{AB}\parallel \overline{FD}$ and $\overline{HE}$ is a transversal, so by theorem 3-2,

  $\angle DCE\cong \angle BAE$ (Corresponding angles)

Thus,

  $m\angle DCE=m\angle BAE=60$

Similarly, lines $\overline{HE}\parallel \overline{FG}$ and $\overline{FD}$ is a transversal, so by theorem 3-2,

  $\angle DFG\cong \angle DCE$ (Corresponding angles)

Thus,

  $\begin{array}{c}m\angle DFG=m\angle DCE\\ 3y+6=60\\ 3y=60-6\end{array}$

  $\begin{array}{c}3y=54\\ y=\frac{54}{3}\\ y=18\end{array}$