# 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}

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