Chapter 5.1, Problem 6WE

To determine

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

Answer to Problem 6WE

  $a=8$

  $b=15$

  $x=80°$

  $y=70°$

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,

  $a=8$

And

  $b=15$

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

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

That is,

  $m\angle x=m\angle QPS=80$

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

  $m\angle PQR+m\angle QRS=180$   (Supplementary angles)

  $\begin{array}{c}\left(30+y\right)+x=180\\ 30+y+80=180\\ 110+y=180\end{array}$

  $\begin{array}{c}y=180-110\\ y=70\end{array}$