Question

Find the measure of y in the given figure.

• A. ${30}^{\circ }$
• B. ${60}^{\circ }$
• C. ${80}^{\circ }$
• D. ${45}^{\circ }$

Answer 1

The correct option is A ${30}^{\circ }$

Given: PQ = QR = RP = RS

Here, $△PQR$ is an equilateral triangle because all sides are equal in length.
Therefore, each angle measures 60${}^{\circ }$.

Since $\angle PRS$ is an exterior angle to $△PQR$,
$\angle PQR+\angle QPR=\angle PRS=x$
$\Rightarrow x={60}^{\circ }+{60}^{\circ }$
$\Rightarrow x={120}^{\circ }$

In $△$ PRS,
RS = PR (Given)
$\therefore △$ PRS is an isosceles triangle.
$\Rightarrow \angle RPS=\angle RSP$
(Base angles of an isosceles triangle)

In $△PRS$,
$\angle PRS+\angle RSP+\angle SPR={180}^{\circ }$
(Sum of all the interior angles of a triangle is ${180}^{\circ }$)
$\Rightarrow x+y+y={180}^{\circ }$
$\Rightarrow x+2y={180}^{\circ }$
$\Rightarrow 2y={180}^{\circ }-{120}^{\circ }$
$\Rightarrow 2y={60}^{\circ }$
$\Rightarrow y={30}^{\circ }$