Ex.3.3 Q12 Understanding Quadrilaterals Solution-Ncert Maths Class 8

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Question

 Find the measure of \(\angle {\rm{P }}\) and \(\angle {\rm{S}}\) if \(SP\) is parallel to \(RQ\) in Fig .

(If you find \({\rm{m}}\angle {\rm{R}}\), is there more than one method to find \({\rm{m}}\angle {\rm{P}}\,?\)

Text Solution

What is Known?

Given figure is a Quadrilateral.

What is Unknown?

Find \({\rm{m}}\angle P{\rm{ and m}}\angle S\)

Reasoning:

Sum of the measures of all the interior angles of a quadrilateral is \(360^\circ\).

Steps:

Given \(SP\) is parallel \(RQ\) and \(SR\) is the traversal drawn to these lines. Hence,

\[\begin{align}\angle \text{S}+\angle \text{R}&={{180}^{\text{o}}}\\\angle \text{S}+{{90}^{\circ}}&={{180}^{\circ}}  \\\angle \text{S}&={{180}^{\circ}}-{{90}^{\circ}}  \\\angle \text{S}&={{90}^{\circ}}  \\
\end{align}\]

Using the angle sum property of a quadrilateral,

\[\begin{align}\angle \text{S}+\angle \text{P}+\angle \text{Q}+\angle \text{R}&={{360}^{\text{o}}}  \\{{90}^{\text{o}}}+\angle \text{P}+{{130}^{\text{o}}}+{{90}^{\text{o}}}&={{360}^{\text{o}}}  \\
\angle \text{P}+{{310}^{\text{o}}}&={{360}^{\text{o}}}  \\\angle \text{P}&={{360}^{\text{o}}}-{{310}^{\text{o}}}  \\\angle \text{P}&={{50}^{\text{o}}}  \\\end{align}\]