Question

In fig 5.23, $\square$PQRS and $\square$ABCR are parallelogram if $\angle$P = 110$°$, find the measures of all the angles of the parallelogram ABCR.

Answer 1

Consider the figure given in the question of two parallelograms $\square$PQRS and $\square$ABCR
Here,
$\square$PQRS is a parallelogram. [Given]
And,
$\angle$P and $\angle$Q are adjacent angles of the parallelogram.
Now,
$\angle$P + $\angle$Q = 180° [Adjacent angles of a parallelogram are supplementary.]
$\Rightarrow$110° + $\angle$Q = 180°
$\Rightarrow$$\angle$Q = 180° – 110°
$\Rightarrow$$\angle$Q = 70°
Now,
$\angle$P = $\angle$R [Opposite angles of a parallelogram are equal.]
$\angle$P = 110°
Thus, we have:
$\angle$R = 110°
And,
$\angle$Q = $\angle$S [Opposite angles of a parallelogram are equal]
Also,
$\angle$Q = 70°
Thus, we have:
$\angle$S = 70°
$$\begin{gathered} \mathrm{In}\mathrm{parallelogram}\square \mathrm{PQRS} \\ \mathrm{PQ}\parallel \mathrm{SR}\left[\mathrm{Opposite}\mathrm{sides}\mathrm{of}\mathrm{parallelogram}\mathrm{are}\mathrm{parallel}\right] \\ \mathrm{or},\mathrm{PQ}\parallel \mathrm{AR} \\ \mathrm{SP}\parallel \mathrm{RQ}\left[\mathrm{Opposite}\mathrm{sides}\mathrm{of}\mathrm{parallelogram}\mathrm{are}\mathrm{parallel}\right] \\ \mathrm{or},\mathrm{SP}\parallel \mathrm{AB} \end{gathered}$$

Thus, we get:
$\angle$A = $\angle$S = 70° [Corresponding angles]
$\angle$A = $\angle$C = 70° [Opposite angles of a parallelogram are equal]
$\angle$B = $\angle$R = 110° [Opposite angles of a parallelogram are equal]