Chapter 5.1, Problem 8WE

To determine

To find: the values of a, b, x and y.

Answer to Problem 8WE

  $a=9$

  $b=11$

  $x=33°$

  $y=27°$

Explanation of Solution

Given:

The diagram:

Consider the figure shown below:

Here it is given that $PQRS$ is a parallelogram.

Now, by theorem 5-3, $\overline{PR}$ and $\overline{QS}$ bisect each other.

That is, O is the midpoint of $\overline{PR}$ and $\overline{QS}$ .

Thus,

  $\begin{array}{c}PO=OR\\ b=11\end{array}$

And

  $\begin{array}{c}QO=OS\\ a=9\end{array}$

Now, both pair of opposite sides of a parallelogram are parallel. So, $\overline{PQ}\parallel \overline{SR}$ and $\overline{QS}$ is a transversal, so by theorem 3-2,

  $\angle PQS\cong \angle QSR$   (Alternate interior angles)

That is,

  $\begin{array}{c}m\angle PQS=m\angle QSR\\ x=33\end{array}$

Now as $\overline{PQ}\parallel \overline{SR}$ and $\overline{PS}$ is a transversal, so by theorem 3-3,

  $m\angle QPS+m\angle PSR=180$   (Supplementary angles)

  $\begin{array}{c}\left(y+50\right)+\left(70+33\right)=180\\ y+50+103=180\\ y+153=180\end{array}$

  $\begin{array}{c}y=180-153\\ y=27\end{array}$