Ex.3.3 Q2 Understanding Quadrilaterals Solution-Ncert Maths Class 8

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Question

Consider the following parallelograms. Find the values of the unknowns \(x\), \(y\), \(z\).

Text Solution

i) What is Known?

\(ABCD\) is a parallelogram.

What is Unknown?

Values of \(x\), \(y\), \(z\).

Reasoning:

In a parallelogram, opposite angles are equal and adjacent angles are supplementary. Using this property, we can calculate the measure of unknown angles.

Steps:

Since \( D\) is opposite to \(B\).

So, \({\rm{y }} = {\rm{ 1}}{00^{\rm{o}}}\) [Since opposite angles of a parallelogram are equal]

\(\angle C + \angle {\rm{B}} = {\rm{18}}0^\circ \)(The adjacent angles in a parallelogram are supplementary)

\({\rm{x }} + {\rm{ 1}}00^\circ {\rm{ }} = {\rm{ 18}}0^\circ {\rm{ }}\) (The adjacent angles in a parallelogram are supplementary)

Therefore

\[\begin{align} x  & ={{180}^{{}^\circ }}-{{100}^{{}^\circ }}  \\   {} & ={{80}^{{}^\circ }}  \\\end{align}\]

\({\rm{x}} = {\rm{ z}} = {\rm{8}}0^\circ {\rm{ }}\) [Since opposite angles of a parallelogram are equal]

ii) What is Known?

Given figure is a parallelogram.

What is Unknown?

values of \(x\), \(y\), \(z\).

Reasoning:

In a parallelogram, opposite angles are equal and adjacent angles are supplementary. Using this property, we can calculate the measure of the unknown angles.

Steps:

\[\begin{align}{{\text{x}} + {\rm{5}}{0^\circ }}&{ = {\rm{18}}{0^\circ }\left( \begin{array}{l}{\text{The adjacent angles}}\\{\text{ in a parallelogram }}\\{\text{are supplementary}}\end{array} \right)}\\
{\rm{x}}&{ = {\rm{18}}{0^\circ } - {\rm{5}}{0^\circ }}\\&{ = {\rm{13}}{0^\circ }}\end{align}\]

\[\begin{align} & \text{x }\!=\!\text{y}\!=\!\text{13}0^\circ \left( \begin{array}{l}{\text{Since opposite angles}}\\{\text{ of a parallelogram }}\\{\text{are equal}}\end{array} \right) \\& \text{x}=\text{z}=\text{13}0^\circ \text{(Corresponding angles)} \\\end{align}\]

iii) What is Known?

Given figure is a parallelogram.

What is Unknown?

values of \(x\), \(y\), \(z\).

Reasoning:

In a parallelogram, opposite angles are equal and adjacent angles are supplementary. Using this property, we can calculate the measure of the unknown angles

Steps:

\[\begin{align}z&={{80}^{\text{o}}}\text{ (Corresponding angles) }  \\y&={{80}^{\text{o}}}\left( \begin{array}{l}{\text{since opposite angles}}\\{\text{ of a parallelogram }}\\{\text{are equal}}\end{array} \right) \\x+y&={{180}^{\text{o}}}\left( \begin{array}{l}{\text{Adjacent angles }}\\{\text{are supplementary}}\end{array} \right)\\x+{{80}^{\text{o}}}&={{180}^{\text{o}}}  \\x&={{180}^{\text{o}}}-{{80}^{\text{o}}}  \\x&={{100}^{{}^\circ }}  \\\end{align}\]

\[\text{Therefore x}={{100}^{{}^\circ }},\,\,\,\text{y}={{80}^{{}^\circ }},\,\,\,\text{z}={{80}^{{}^\circ }}\]

iv) What is the known?

Given figure is a parallelogram.

What is unknown?

Values of \(x\), \(y\), \(z\).

Reasoning:

In a parallelogram, opposite angles are equal and adjacent angles are supplementary. Using this property, we can calculate the measure of the unknown angles.

Steps:

 

\[\begin{align}\text{x}+\text{y}+{{30}^{\text{o}}}&={{180}^{\text{o}}}\left( \begin{array}{l}{\text{Angle sum}}\\{\text{ property of }}\\{\text{triangles}}\end{array} \right)\\ 
\text{x}&\!\!=\!\!{{90}^{\text{o}}}\left( \begin{array}{l}{\text{Vertically  }}\\{\text{opposite angles}}\end{array} \right)\\ {{90}^{{}^\circ }}+\text{y}+{{30}^{\text{o}}}&={{180}^{\text{o}}} \\ \text{y}+120&={{180}^{\text{o}}} \\ \text{y}&={{180}^{\text{o}}} \\ \text{z}&={{60}^{\text{o}}}\end{align}\]

Therefore

\[\text{z}=\text{y}={{60}^{\text{o}}}\left( \begin{array}{l}{\text{Alternate interior }}\\{\text{angles are equal}}\end{array} \right)\]

v) What is Known?

Given figure is a parallelogram.

What is Unknown?

Values of \(x\), \(y\), \(z\).

Reasoning:

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

Using this property, we can calculate the unknown angles.

Steps:

\[\begin{align}y&=112^\circ \end{align}\]

(Since opposite angles of a parallelogram  are equal)

\[\begin{align}x+y+40^\circ=180^\circ \end{align}\]

(Angle sum property of triangles)

\[\begin{align}{x}+{{112}^{\circ}}+{{40}^{\circ}}&={{180}^{\circ}} \\{x}+{{152}^{\circ}}&={{180}^{\circ}} \\ {x}&={{180}^{\circ}}-{{152}^{\circ}} \\ {x}&={{28}^{\circ}} \\{z}&={x}={{28}^{\circ}} \end{align}\]

(Alternate interior angles)

Therefore 

\[\begin{align}x &={{28}^{{}^\circ }},{y}={{112}^{{}^\circ }},{z}={{28}^{{}^\circ }} \\\end{align}\]