91Ó°ÊÓ

Diagonals of a parallelogram bisect each other.

In the given figure, $\stackrel{¯}{PR}$and $\stackrel{¯}{QS}$ bisect each other and O is the midpoint of $\stackrel{¯}{PR}$and $\stackrel{¯}{QS}$implies that

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

The opposite sides of a parallelogram are congruent.

In the given figure, $\stackrel{¯}{PS}â‰\dots \stackrel{¯}{QR}$implies that $b=8$.

Both pairs of opposite sides of a parallelogram are parallel, that is, $\stackrel{¯}{PS}âˆ\yen \stackrel{¯}{QR}$and $\stackrel{¯}{QS}$is a transversal then angle made by a transversal with respect to two lines are equal.$∠PSQâ‰\dots ∠SQR$

Therefore,

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

Consider $\mathrm{Δ}QOR$, here $\stackrel{¯}{OR}â‰\dots \stackrel{¯}{QR}$.

Therefore, by isosceles triangle theorem, $∠OQRâ‰\dots ∠ROQ$, that is,

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

Consider $\mathrm{Δ}QOR$,

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

Therefore,

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

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