Chapter 5.1, Problem 7WE

To determine

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

Answer to Problem 7WE

  $a=5$

  $b=3$

  $x=120°$

  $y=22°$

Explanation of Solution

Given:

The diagram:

Consider the figure shown below:

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

So, by theorem 5-1, its opposite sides are congruent.

Here $\overline{PS}$ , $\overline{QR}$ and $\overline{PQ}$ , $\overline{SR}$ are the opposite sides of the parallelogram $PQRS$ . So,

$\overline{PS}\cong \overline{QR}$ And $\overline{PQ}\cong \overline{SR}$

Thus,

  $a=5$

And

  $b=3$

Also by theorem 5-2, opposite angle of parallelogram $PQRS$ are congruent. So,

$\angle x\cong \angle PSR$   (Opposite angles of parallelogram $ABCD$ )

That is,

  $m\angle x=m\angle PSR=120$

Now apply theorem 3-11 on the triangle $△PQR$ ,

  $\begin{array}{c}m\angle PQR+m\angle QRP+m\angle RPQ=180\\ 120+m\angle QRP+38=180\\ m\angle QRP+158=180\end{array}$

  $\begin{array}{c}m\angle QRP=180-158\\ m\angle QRP=22\end{array}$

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

  $\angle SPR\cong \angle QRP$   (Alternate interior angles)

That is,

  $\begin{array}{c}m\angle SPR=m\angle QRP\\ y=22\end{array}$