Chapter 3.2, Problem 11WE

To determine

To find: the values of x and y .

Answer to Problem 11WE

  $x=14$

  $y=9$

Explanation of Solution

Given:

The diagram:

Concept Used:

Theorem 3-2: If two parallel lines are cut by a transversal then the alternate interior 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:

In the above figure, lines $\overline{AB}\parallel \overline{XY}$ and $\overline{XB}$ is a transversal, so by theorem 3-2,

  $\angle YXB\cong \angle XBA$ (Alternate interior angles)

That is,

  $m\angle YXB=m\angle XAB$

That is,

  $\begin{array}{c}3x=42\\ x=\frac{42}{3}\\ x=14\end{array}$

Now, lines $\overline{AB}\parallel \overline{XY}$ and $\overline{YB}$ is a transversal, so by theorem 3-3,

  $m\angle XYB+m\angle YBA=180$ (Same-side interior angles)

But $\angle YBA=\angle YBX+\angle XBA$ , so,

  $m\angle YBA=m\angle YBX+m\angle XBA$

This implies that,

  $\begin{array}{c}m\angle XYB+m\angle YBA=180\\ m\angle XYB+m\angle YBX+m\angle XBA=180\\ 90+\left(6y-6\right)+42=180\\ 90-6+42+6y=180\end{array}$

  $\begin{array}{c}126+6y=180\\ 6y=180-126\\ 6y=54\\ y=9\end{array}$