Question

In Exercises 5-10 quad. PQRS is a parallelogram. Find the values of a, b, x and y.

Short answer

The values of a, b, x and y are $9,11,33°$and$27°$ respectively.

Step-by-step solution

Step 1. Consider a following figure.

Step 2. Apply property of parallelogram.

Diagonals of a parallelogram bisect each other.

Step 3. Description of step.

In the given figure, $\overline{PR}$and role="math" localid="1637641507892" $\overline{QS}$ bisect each other and O is the midpoint of $\overline{PR}$and $\overline{QS}$implies that

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

And

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

Step 4. Description of step.

Both pairs of opposite sides of a parallelogram are parallel, that is, $\overline{PQ}\left|\right|\overline{SR}$and $\overline{SQ}$is a transversal then angle made by a transversal with respect to two lines are equal.

$\angle PQS\cong \angle QSR$

Therefore

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

Step 5. Description of step.

Both pairs of opposite sides of a parallelogram are parallel, that is, $\overline{PQ}\left|\right|\overline{SR}$and $\overline{PS}$is a transversal then same side interior angles are supplementary.

$\begin{array}{c}m\angle QPS+m\angle PSR=180\\ 50+y+70+33=180\\ y=180-153\\ =27°\end{array}$

The values of a, b, x and y are $9,11,33°$and$27°$ respectively.