Question

1. there is a quadrilateral pqrs, with angle at q being 63°, angle at p being 117°, angle at r being 117°, and angle at s being unknown (marked as ?).

Explanation:

Step1: Recall the sum of interior angles of a quadrilateral

The sum of the interior angles of any quadrilateral is \( (4 - 2)\times180^{\circ}= 360^{\circ} \).

Step2: Let the measure of angle \( S \) be \( x \)

We know the measures of angles \( Q = 63^{\circ} \), \( P = 117^{\circ} \), \( R = 117^{\circ} \), and angle \( S=x \). So we can set up the equation: \( 63^{\circ}+ 117^{\circ}+ 117^{\circ}+x=360^{\circ} \).

Step3: Simplify the left - hand side of the equation

First, calculate the sum of the known angles: \( 63^{\circ}+ 117^{\circ}+ 117^{\circ}=63^{\circ}+(117^{\circ}+ 117^{\circ})=63^{\circ}+234^{\circ}=297^{\circ} \).

Step4: Solve for \( x \)

We have the equation \( 297^{\circ}+x = 360^{\circ} \). Subtract \( 297^{\circ} \) from both sides of the equation: \( x=360^{\circ}- 297^{\circ}=63^{\circ} \).

Answer:

The measure of angle \( S \) is \( 63^{\circ} \)