Exercise 11.1 (Solutions)

Q.1 One angle of parallelogram is $130^\circ$. Find the measures of its remaining angles.

Solution:

In a parallelogram $ABCD$, $m\angle B=130^\circ$

$m\angle B=m\angle D$ (Opposite angles of parallelogram)

$m\angle B=m\angle D=130^\circ$

We know that
$$\begin{align} & m\angle A +\,\,m\angle B=180^\circ \\ & m\angle A+\,{{130}^{\circ }}={{180}^{\circ }}\\ & m\angle A={{180}^{\circ }}-{{130}^{\circ }}\\ & m\angle A={{50}^{\circ }}\end{align}$$ Also we have $m\angle A=m\angle C$ (Opposite angles of parallelogram)
$\Rightarrow \,\,\,\,m\angle C=50^\circ$

Q.2 One exterior angle formed on producing one side of a parallelogram is $40^\circ$. Find the measures of its interior angles.

Solution:
Given In a parallelogram $ABCD$, $m\angle DAM=40^\circ$

To find: $m\angle B=?$, $m\angle DAB=?$, $m\angle C=?$, $m\angle D=?$

$$\begin{align} & m\angle DAM+m\angle DAB=180^\circ \\ & 40^\circ + m\angle DAB= 180^\circ \\ & m\angle DAB=180^\circ- 40^\circ \\ & m\angle DAB=140^\circ \end{align}$$ Also $$\begin{align} & m\angle DAB+m\angle B=180^\circ \\ & 140^\circ+m\angle B=180^\circ \\ & m\angle B=180^\circ-140^\circ \\ & m\angle B=40^\circ \end{align}$$

Now \begin{align} & m\angle B =m\angle D=40^\circ \\ \Rightarrow \,\,\,\,\,\,\,\,\, & m\angle D=40^\circ \end{align}

Also \begin{align} & m\angle C=m\angle DAB \\ \Rightarrow \,\,\,\,\,\,\,\,\, & m\angle C=140^\circ \end{align}