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)

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

\({\rm{x}} = {\rm{ z}} = {\rm{8}}0^\circ {\rm{ }}\) [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:

 

 

 

 

 

 

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

\({\text{x }} = {\rm{ y }} = {\rm{13}}0^\circ {\rm{ }} \text{(Since opposite angles of a parallelogram are equal)}\\{\text{x }} = {\rm{ z }} = {\rm{13}}0^\circ {\rm{ }} \text{(Corresponding angles)}\)

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:

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

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

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:

 

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

\[\text{Therefore z}=\text{y}={{60}^{\text{o}}}\text{ (Alternate interior angles are equal)}\]

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:

\[\begin{align}{y}&={{112}^{\circ}} \text{ (Since opposite angles of a parallelogram are equal) }  \\
{x}+{y}+{{40}^{\circ}}&={{180}^{\circ}}\text{ (Angle sum property of triangles) }  \\{x}+{{112}^{\circ}}+{{40}^{\circ}}&={{180}^{\circ}}  \\{x}+{{152}^{\circ}}&={{180}^{\circ}}  \\ {x}&={{180}^{\circ}}-{{152}^{\circ}} \\ {x}&={{28}^{\circ}}  \\{z}&={x}={{28}^{\circ}}  \\ & \text{ (Alternate interior angles) }  \\\text{ Therefore }x &={{28}^{{}^\circ }},{y}={{112}^{{}^\circ }},{z}={{28}^{{}^\circ }}  \\\end{align}\]