Question

Question 164
Two angles of a quadrilateral are each of measure ${75}^{\circ }$ and the other two angles are equal. What is the measure of these two angles? Name the possible figures so formed.

Answer 1

Let ABCD be a quadrilateral,
where $\angle A=\angle C={75}^{\circ }$ and $\angle B=\angle D=x$ [say]

Then, by the angle sum property of a quadrilateral, we have
$\angle A+\angle B+\angle C+\angle D={360}^{\circ }\Rightarrow {75}^{\circ }+x+{75}^{\circ }+x={360}^{\circ }\Rightarrow 2x={360}^{\circ }-{150}^{\circ }\Rightarrow 2x={210}^{\circ }\Rightarrow x={105}^{\circ }$
Thus, other two angles are of ${105}^{\circ }$ each.
Since, opposite angles are equal, therefore the quadrilateral is a parallelogram.