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

Answer :

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 $$m∠P = 50^{\circ}$$