12.Find the measure of ∠P and ∠S if \(\bar { SP } ∥ \bar { RQ }\) in figure, is there any other method to find m∠P?)

We have, \(∠Q = 130° and ∠R = 90^{\circ}\) and \(\bar { SP } || \bar { RQ }\)
\(∠P + ∠Q = 180^{\circ}\quad\) [Adjacent angles]

\(\Rightarrow ∠P + 130^{\circ} = 180^{\circ}\)

\(\Rightarrow ∠P = 180^{\circ} – 130^{\circ} = 50^{\circ}\)

and also we have, \(∠S + ∠R = 180^{\circ}\quad\) [Adjacent angles]

\(\Rightarrow ∠S + 90^{\circ} = 180^{\circ}\)

\(\Rightarrow ∠S = 180^{\circ} – 90^{\circ} = 90^{\circ}\)
Hence, \(m∠P = 50^{\circ} \;and \; m∠S = 90^{\circ}\)
Alternate Method:
\(∠Q = 130^{\circ}, ∠R = 90^{\circ} and ∠S = 90^{\circ}\)

We know that

\(∠P + ∠Q + ∠R + ∠Q = 360^{\circ}\quad\) [Angle sum property of quadrilateral]

\(\Rightarrow ∠P + 130^{\circ}+ 90^{\circ} + 90^{\circ} = 360^{\circ}\)

\(\Rightarrow ∠P + 310^{\circ} = 360^{\circ}\)

\(\Rightarrow ∠P = 360^{\circ}– 310° = 50^{\circ}\)

Hence \(m∠P = 50^{\circ}\)