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 $8,8,56°$and$68°$ 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 $\overline{QS}$ bisect each other and O is the midpoint of $\overline{PR}$and $\overline{QS}$implies that

$\begin{array}{c}PO=OR\\ a=8\end{array}$

Step 4. Apply property of parallelogram.

The opposite sides of a parallelogram are congruent.

In the given figure, $\overline{PS}\cong \overline{QR}$implies that $b=8$.

Step 5. Description of step.

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

Therefore,

$\begin{array}{c}m\angle SQR=m\angle PSQ\\ =56°\end{array}$

Step 6. Description of step.

Consider $\Delta QOR$, here $\overline{OR}\cong \overline{QR}$.

Therefore, by isosceles triangle theorem, $\angle OQR\cong \angle ROQ$, that is,

$\begin{array}{c}m\angle ROQ=m\angle OQR\\ x=56\end{array}$

Step 7. Description of step.

Consider $\Delta QOR$,

The sum of the measures of all interior angles of a triangle is 180.

Therefore,

$\begin{array}{c}m\angle QRO+m\angle ROQ+m\angle OQR=180\\ y+56+56=180\\ y=180-112\\ =68°\end{array}$

The values of a, b, x and y are $8,8,56°$and$68°$ respectively.