Question

Look at the figure given below. Can you find the measure of $\angle y$ in the figure?

• A. ${30}^{\circ }$
• B. ${60}^{\circ }$
• C. ${90}^{\circ }$
• D. ${120}^{\circ }$

Answer 1

The correct option is A

${30}^{\circ }$

Given: PQ = QR = RP = RS
$△PQR$ is an equilateral triangle because all sides are equal in length.
Since we know that the sum of all the angles in a triangle is 180${}^{\circ }$,
Therefore, each angle measures 60${}^{\circ }$.

Now, using exterior angle property,
$\angle PQR$ + $\angle QPR$ = $\angle PSR=x$
$\Rightarrow x={60}^{\circ }+{60}^{\circ }$
$\Rightarrow x={120}^{\circ }$
In $△$ PRS,
RS = PR (Given)
$\therefore △$ PRS is an isosceles triangle.

Using angle sum property,
$\angle PRS+\angle RSP+\angle SPR={180}^{\circ }$
$\Rightarrow x+y+y={180}^{\circ }$
$\Rightarrow x+2y={180}^{\circ }$
$\Rightarrow 2y={180}^{\circ }-{120}^{\circ }$
$\Rightarrow 2y={60}^{\circ }$
$\Rightarrow y={30}^{\circ }$