Ex.3.3 Q6 Understanding Quadrilaterals Solution-Ncert Maths Class 8

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Question

Two adjacent angles of a parallelogram have equal measure. Find the measure of each of the angles of the parallelogram.

Text Solution

What is Known?

Two adjacent angles of a parallelogram have equal measure.

What is Unknown?

Measure of each of the angles of the parallelogram.

Reasoning:

In parallelogram opposite angles are equal and adjacent angles are supplementary.

Using this property, we can calculate the unknown angles.

Steps:

In parallelogram \(ABCD\),

\(\angle {\rm{A }}\) and \(\angle {\rm{D}}\) are supplementary since \(DC\) is parallel to \(AB\) and with transversal \( DA\),

making \(\angle {\rm{A}}\) and \(\angle D\) interior opposite.

\(\angle {\rm{A }}\) and \(\angle {\rm{B }}\) are also supplementary since \(AD\) is parallel to \( BC\) and with transversal \(BA\), making \(\angle {\rm{A}}\) and \(\angle {\rm{B}}\) interior opposite.

Sum of adjacent angles \(= {\rm{18}}0^\circ \)

Let each adjacent angle be \(x\)

Since the adjacent angles in a parallelogram are supplementary.

\[\begin{align}x + x &= 180^\circ \\2x &= 180^\circ \\x &= \frac{{{\rm{18}}0^\circ }}{2}\end{align}\]

Hence, each adjacent angle is \(90\).

\[\begin{align}\angle \text{A }&=\angle \text{B}=\text{ 9}0{}^\text{o}\text{ }\left( \text{adjacent angles} \right) \\ \angle \text{C }&=\angle \text{A}=\text{ 9}0{}^\text{o}\text{ }\left( \text{Opposite angles} \right) \\ \angle \text{D}&=\angle \text{B}=\text{9}0{}^\text{o}\left( \text{Opposite angles} \right)  \end{align}\]

Thus, each angle of the parallelogram measures \(90^\circ\).