Question

If an angle of a parallelogram is two-third of its adjacent angle, find the angles of the parallelogram.

Answer 1

Let in a parallelogram $ABCD$,

$\angle A=x$

Then its adjacent angle , $\angle B=\frac{2}{3}x$

But, $\angle A+\angle B={180}^{\circ }$ [Since, sum of the consecutive angles of a parallelogram is ${180}^{\circ }$.

$\Rightarrow x+\frac{2}{3}x={180}^{\circ }$

$\Rightarrow \frac{3x+2x}{3}={180}^{\circ }$

$\Rightarrow \frac{5x}{3}={180}^{\circ }$

$\Rightarrow x={180}^{\circ }\times \frac{3}{5}$

$\Rightarrow x={36}^{\circ }\times 3$

$\Rightarrow x={108}^{\circ }$

$\Rightarrow \angle A={108}^{\circ }$---(1)

$\Rightarrow \angle B={180}^{\circ }-{108}^{\circ }={72}^{\circ }$---(2)

Since, opposite angles of a parallelogram are equal.

$\Rightarrow \angle A=\angle C={108}^{\circ }$ ---(3)

and $\Rightarrow \angle B=\angle D={72}^{\circ }$---(4)

Hence angles are ${108}^{\circ },{72}^{\circ },{108}^{\circ },{72}^{\circ }$.