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 $5,3,120°$and $22°$respectively.

Step-by-step solution

Step 1. Apply property of parallelogram.

The opposite sides of a parallelogram are congruent.

Step 2. Description of step.

In the given figure, $\overline{PS}\cong \overline{QR}$implies that $a=5$and $\overline{PQ}\cong \overline{SR}$implies that $b=3$.

Step 3. Apply property of parallelogram.

The opposite angles of a parallelogram are congruent.

Step 4. Description of step.

From the given figure, it can be observed that $\angle x\cong \angle PSR$implies that

$\begin{array}{c}x=m\angle PSR\\ =120°\end{array}$

Step 5. Description of step.

Consider $\Delta PQR$,

The sum of measures of all angles of a triangle is 180. Therefore,

$\begin{array}{c}m\angle QPR+m\angle PQR+m\angle QRP=180\\ 38+120+m\angle QRP=180\\ m\angle QRP=180-158\\ =22°\end{array}$

Step 6. Description of step.

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

$\angle SPR\cong \angle QRP$

Therefore,

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

Therefore, the values of a, b, x and y are $5,3,120°$and $22°$respectively.