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 $10,4,90°$and$45°$ respectively.

Step-by-step solution

Step 1. Consider the following figure.

Step 2. Apply property of parallelogram.

The opposite sides of a parallelogram are congruent.

In the given figure, $\overline{PS}\cong \overline{QR}$implies that $a=8$and $\overline{PQ}\cong \overline{SR}$implies that $b=15$.

Thus,

$\begin{array}{c}PQ=SR\\ PQ=SO+SR\\ a=6+4\\ =10\end{array}$

And$PS=QR$

Step 3. Description of step.

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

From the figure, it can be observed that the transversal $\overline{SR}$is perpendicular to the line $\overline{PS}$, then $\overline{SR}$ is perpendicular to the line $\overline{QR}$. That is,

$\begin{array}{c}m\angle SRQ=90\\ x=90°\end{array}$

Step 4. Description of step.

Consider $\Delta QOR$, the sum of the measures of all the angles of a triangle is 180.

Therefore,

$\begin{array}{c}m\angle QRO+m\angle ROQ+m\angle OQR=180\\ 90+45+y=180\\ 135+y=180\\ y=45\end{array}$

Step 5. Description of step.

Consider $\Delta QOR$, here $\angle OQR\cong \angle ROQ$.

Therefore, by converse isosceles triangle theorem, $\overline{OR}\cong \overline{QR}$, that is,

$\begin{array}{c}QR=OR\\ =4\end{array}$

As, $PS=QR$,

$\begin{array}{c}PS=4\\ b=4\end{array}$

Therefore, the values of a, b, x and y are $10,4,90°$ and $45°$ respectively.