# Excercise 3.3 Understanding Quadrilaterals- NCERT Solutions Class 8

Understanding Quadrilaterals

Exercise 3.3

## Question 1

Given a parallelogram \(ABCD\). Complete each statement along with the definition or property used.

(i)\(\, AD\, = \,\_\_\_\_\_\_\_\_\_\)

(ii)\(\, \angle DCB\, = \,\_\_\_\_\_\_\)

(iii)\(\,OC\, = \,\_\_\_\_\_\_\_\_\)

(iv) \(\,m\,\angle DAB\,\!\!+\!\!\,m\,\angle CDA\,\!\!=\_\_\_\_\_\_\_\)

### Solution

**Video Solution**

**What is Known?**

\(ABCD\) is a parallelogram.

**What is Unknown?**

\(\begin{align}&{\rm{AD}},\angle DCB,\,{\rm{OC}},\,\,\\&{\rm{m}}\angle DAB +\!{\rm{m}}\angle CDA\end{align}\)

**Reasoning:**

We can use the properties of parallelogram to determine the solution.

**Steps:**

i) The opposite sides of a parallelogram are of equal length.

\({\rm{AD }} = {\rm{ BC}}\)

(ii) In a parallelogram, opposite angles are equal in measure.

\(\angle {\rm{DCB }} = {\rm{ }}\angle {\rm{DAB}}\)

(iii) In a parallelogram, diagonals bisect each other. Hence,

\({\rm{OC }} = {\rm{ OA}}\)

(iv)In a parallelogram, adjacent angles are supplementary to each other. Hence,

\(m\angle {\rm{DAB }} + {\rm{ m}}\angle {\rm{CDA }} = {\rm{18}}0^\circ \)

## Question 2

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

### Solution

**Video 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, \({{y }} = {\rm{ 1}}{00^{\rm{o}}}\) [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)

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

Therefore,

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

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

\(y=112^\circ\)(Since opposite angles of a parallelogram are equal)

\[\begin{align}x+y+40^\circ&=180^\circ\\\text{(Angle sum }&\text{property of triangles)} \end{align}\]

\[\begin{align}{x}+{{112}^{\circ}}+{{40}^{\circ}}&={{180}^{\circ}} \\{x}+{{152}^{\circ}}&={{180}^{\circ}} \\ {x}&={{180}^{\circ}}-{{152}^{\circ}} \\ {x}&={{28}^{\circ}} \\{z}&={x}={{28}^{\circ}}\\\text{ (Alternate}&\text{ interior angles)}\end{align}\]

Therefore,

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

## Question 3

Can a quadrilateral \(ABCD\) be a parallelogram if

(i) \(∠D +∠B = 180^\circ?\)

(ii) \(AB = DC = 8 \rm \,cm, AD = 4 \rm \,cm \) and \(BC = 4.4 \rm \,cm?\)

(iii) \(∠A = 70^\circ\) and \(∠C = 65^\circ?\)

### Solution

**Video Solution**

i) **What is Known?**

Given figure is a quadrilateral

**What is Unknown?**

If \(ABCD\) is a parallelogram when \(\angle {\rm{D }} + \angle {\rm{B }} = {\rm{18}}0^\circ ?\)

**Reasoning:**

A parallelogram is a quadrilateral whose opposite sides are parallel.

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

Using this property, we can calculate the unknown angles.

**Steps:**

Using the angle sum property of a quadrilateral,

\[\begin{align}\angle A+\angle B+\angle D+\angle C&={{360}^{\circ}} \\\angle A+\angle C+{{180}^{\circ}}&={{360}^{\circ}} \\\angle A+\angle C&={{360}^{\circ}}-{{180}^{\circ}} \\\ \angle A+\angle C&={{180}^{\circ}}\end{align}\]

(Opposite angles should also be of same measures.)

For \(\angle {\rm{D}} + \angle {\rm{B}}\) \(= {\rm{18}}0^\circ \), is a parallelogram.

If the following conditions is fulfilled, then \(ABCD\) is a parallelogram.

The sum of the measures of the adjacent angles should be \(180^\circ\).

Opposite angles should also be of same measure.

ii) **What is Known?**

Given figure is a quadrilateral.

**What is Unknown?**

\(ABCD\) be a parallelogram if \(AB = DC = 8 \rm \,cm, AD = 4 \rm \,cm\) and \(BC = 4.4 \rm \,cm\)

**Reasoning:**

A parallelogram is a quadrilateral whose opposite sides are parallel.

**Steps:**

**Property of parallelogram:** The opposite sides of a parallelogram are of equal length. Opposite sides \(AD\) and \(BC \) are of different lengths. So, it’s not parallelogram.

iii) **What is Known?**

Given figure is a quadrilateral.

**What is Unknown?**

\(ABCD\) be a parallelogram if \(\angle {\rm{A }} = {\rm{ 7}}0^\circ {\rm{ }}\) and \(\angle {\rm{C }} = {\rm{ 65}}^\circ ?\)

**Reasoning:**

A parallelogram is a quadrilateral whose opposite sides and angles are equal.

**Steps:**

**Property:** In a parallelogram opposite angles are equal.

So, \(\angle {\rm{A }} = {\rm{ 7}}0^\circ {\rm{ }}\)and \( \angle {\rm{C }} = {\rm{ 65}}^\circ {\rm{ }}\)are not equal.

So \(ABCD\) is not parallelogram.

## Question 4

Draw a rough figure of a quadrilateral that is not a parallelogram but has exactly two opposite angles of equal measure.

### Solution

**Video Solution**

**What is Known?**

Draw a figure of quadrilateral having two opposite angles of equal measure.

**What is Unknown?**

\(ABCD\) is quadrilateral whose opposite angles are equal.

**Reasoning:**

The opposite angles of a parallelogram are equal.

**Steps:**

In a kite, the angle between unequal sides are equal.

Draw line from \(A\) to \(C\) and we will get two triangles with common base \(AC\).

In \(∆ABC\) and \(∆ADC \) we have,

\({\rm{AB}} = {\rm{ AD }},{\rm{ BC}} = {\rm{CD}}\) ; \(AC\) is common to both

\(\Delta {\rm{ABC }}\!\!\cong\!\!\Delta {\rm{ADC }}\text{[congruent triangles]}\)

Hence corresponding parts of congruent triangles are equal.

Therefore\(\angle {\rm{B}} = \angle {\rm{D}}\)

However, the quadrilateral \(ABCD\) is not a parallelogram as the measures of the remaining pair of opposite angles, \(\angle {\rm{A}}\) and \(\angle {\rm{C}}\), are not equal. Since they form angle between equal sides.

## Question 5

The measures of two adjacent angles of a parallelogram are in the ratio \({\rm{3}}:{\rm{2}}\). Find the measure of each of the angles of the parallelogram.

### Solution

**Video Solution**

**What is Known?**

Given figure is a parallelogram and two adjacent angles are having ratio of \({\rm{3}}:{\rm{2}}\) quadrilateral.

**What is Unknown?**

Measure of Each angles of parallelogram.

**Reasoning:**

A parallelogram is a quadrilateral whose opposite angles are equal.

**Steps:**

We know that the sum of the measures of adjacent angles is \(180º \) for a parallelogram.

\[\begin{align}\angle A + \angle B &= {180^{\rm{o}}}\\3x + 2x &= {180^{\rm{o}}}\\5x &= {180^{\rm{o}}}\\x &= \frac{{{{180}^{\rm{o}}}}}{5}\\x &= \,{36^{\rm{o}}}\end{align}\]

\[\begin{align} \angle A&=\angle C=3x \\ & ={{108}^{\circ}}\text{ }(\text{ Opposite angles }) \\ \angle B&=\angle D=2x \\ & ={{72}^{\circ}}\text{ (Opposite angles) } \end{align}\]

Thus, the measures of the angles of the parallelogram are \(108^\circ, 72^\circ, 108^\circ,\) and \(72^\circ\).

## Question 6

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

### Solution

**Video 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 \(90^\circ\).

## Question 7

The adjacent figure \(HOPE\) is a parallelogram. Find the angle measures \(x, y\) and \( z.\) State the properties you use to find them.

### Solution

**Video Solution**

**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 unknown angles.

**Steps:**

Here,

\[\begin{align}\angle &{\rm{HOP }} + {\rm{ 7}}0^\circ = {\rm{ 18}}0^\circ {\rm{ }}\,\,\\&\left[ {{\text{Angles of linear pair}}} \right]\\\angle &{\rm{HOP}} = {\rm{18}}0^\circ - {\rm{7}}0^\circ \\\angle &{\rm{HOP}} = {\rm{11}}0^\circ \\\angle &{\rm{O }} = \angle E\left( {{\text{opposite angles are equal}}} \right)\\\therefore &{\rm{x}} = {\rm{11}}0^\circ {\rm{ }}\end{align}\]

\[\begin{align}\text{y}&={{40}^{\text{o}}}\\\text{ (Alternate }&\text{interior angles are equal) } \\\text{z}+{{40}^{\circ}}&={{70}^{\circ}}\\&\text{ (Corresponding angles) } \\ \text{z}&={{70}^{\circ}}-{{40}^{\circ}} \\ \text{z}&={{30}^{\circ}} \\ \end{align}\]

\[\therefore {\text{ x }} = {\rm{ 11}}0^\circ ,{\text{ y }} = {\text{ 4}}0^\circ ,{\rm{ z }} = {\text{ 3}}0^\circ \]

## Question 8

The following figures \(GUNS\) and \(RUNS\) are parallelograms. Find \(x\) and \(y\). (Lengths are in \(\rm cm\))

### Solution

**Video Solution**

i) **What is Known?**

Given figure is a parallelogram.

**What is Unknown?**

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

**Reasoning:**

The diagonals of a parallelogram bisect each other, in a parallelogram, the opposite sides have same length.

**Steps:**

In a parallelogram, the opposite sides have same length.

\[\begin{align}{\rm{SG}}\,\rm&= \,{\rm{NU}}\\3x &= 18\\x &= \,\frac{{18}}{3}\\x &= 6\end{align}\]

And,

\[\begin{align}\text{SN}&=\text{GU} \\ 26&=3y-1 \\ 3y&=26+1 \\y&=\frac{27}{3} \\y&=9 \\\end{align}\]

Hence, the measures of \(x\) and \( y\) are \(6 \rm\, cm\) and \(9 \rm \,cm\) respectively.

(ii) **What is Known?**

Given figure is a parallelogram.

**What is Unknown?**

Values of \(x, y\)

**Reasoning:**

The diagonals of a parallelogram bisect each other. In a parallelogram, the opposite sides have same length.

**Steps:**

Property: The diagonals of a parallelogram bisect each other.

\[\begin{align}y{\rm{ }} + {\rm{ }}7{\rm{ }} &= {\rm{ }}20\\y &= 20 - 7\\y{\rm{ }} &= {\rm{ }}13\\

x{\rm{ }} + {\rm{ }}y{\rm{ }} &= {\rm{ }}16\\x{\rm{ }} + {\rm{ }}13{\rm{ }} &= {\rm{ }}16\\x{\rm{ }} &= {\rm{ }}3\end{align}\]

Hence, the measures of \(x\) and \(y\) are \(3 \rm\,cm\) and \(13 \rm\,cm \) respectively.

## Question 9

In the above figure both \(RISK\) and \( CLUE\) are parallelograms. Find the value of \(x\).

### Solution

**Video Solution**

**What is Known?**

In the given figure \(RISK \)and \(CLUE\) are parallelograms.

**What is Unknown?**

Values of \(x\)

**Reasoning:**

The diagonals of a parallelogram bisect each other. Also, in a parallelogram, opposite angles are equal and adjacent angles are supplementary. Using this property, we can calculate the unknown angles.

**Steps:**

In parallelogram \(RISK\)

\({\rm{RKS }} + \angle {\rm{ISK }} = {\rm{ 18}}0^\circ \) (Adjacent angles of a parallelogram are supplementary)

\[\begin{align}{\rm{12}}0^\circ + \,\angle {\rm{ISK }} &= {\rm{18}}0^\circ \\\angle {\rm{ISK }} &= {\rm{ 6}}0^\circ\end{align}\]

\[\begin{align}{\angle {\text{I}}}&{ = \angle {\rm{K}}}\,{\left( \begin{array}{l}{\text{In parallelogram opposite }}\\{\text{angles are equal}}\end{array} \right)}\\&{ = {\rm{12}}{0^\circ }}&\end{align}\]

In parallelogram \(CLUE\)

\[\begin{align}{\angle {\rm{L}}}&{ = \angle {\rm{E}}}\,{\left( \begin{array}{l}{\text{In parallelogram opposite}}\\{\text{ angles are equal}}\end{array} \right)}\\&{ = {\rm{7}}{0^\circ }}&\end{align}\]

The sum of the measures of all the interior angles of a triangle is \(180^\circ.\)

\[\begin{align}x+{{60}^{\circ}}+{{70}^{\circ}}&={{180}^{\circ}} \\x+130&={{180}^{\circ}} \\x&={{180}^{\circ}}-{{130}^{\circ}} \\x&={{50}^{\circ}} \\\end{align}\]

## Question 10

Explain how this figure is a trapezium. Which of its two sides are parallel?

### Solution

**Video Solution**

**What is Known?**

Given figure is a Quadrilateral.

**What is Unknown?**

To identify the two parallel sides of the figure and to prove that it is a trapezium.

**Reasoning:**

Trapezium is a quadrilateral having one pair of parallel sides.

**Steps:**

In the given figure \( KLMN\),

\(\angle {\rm{L }}\,+\) \({\rm{ }}\angle {\rm{M }}\) \(= {\rm{18}}0^\circ \) [two pair of adjacent angles (which form pairs of consecutive interior angles) are supplementary]

\[ = {\rm{ 8}}0^\circ {\rm{ }} + {\rm{1}}00^\circ = {\rm{18}}0^\circ \]

Therefore, \(KN\) is parallel to \(ML\)

Hence, \(KLMN\) is a trapezium as it has a pair of parallel sides: \(KN \) and \(ML\).

## Question 11

Find \({\rm{m}}\,\angle {\rm{C }}\) in Fig \(3.33\) if \(AB \) is parallel to \( DC\).

### Solution

**Video Solution**

**What is Known?**

Given figure is a Quadrilateral with two sides running parallel and one angle is given.

**What is Unknown?**

Find \({\rm{m }}\angle {\rm{C}}\)

**Reasoning:**

Trapezium is a quadrilateral with one pair of parallel sides.

**Steps:**

Given figure \(ABCD\) is a Trapezium, in which \(AB\) is parallel to \( DC\).

Here,

\[\begin{align}\angle {\rm{B}} + \angle {\rm{C}} &= {180^ \circ }\left[ \begin{array}{l}{\text{pair of adjacent }}\\{\text{angles are }}\\{\text{supplementary}}\end{array} \right]\\{\rm{12}}{0^\circ } + \angle {\rm{C}} &= {180^ \circ }\\\angle {\rm{C}} &= {180^ \circ }\\\angle {\rm{C}} &= {180^ \circ } - {120^ \circ }\\\angle {\rm{C }}&={60^ \circ }\\\therefore \,\,m\angle {\rm{C }}&={60^ \circ }\end{align}\]

## Question 12

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 \({\rm{m}}\angle {\rm{P}}\,?\)

### Solution

**Video 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,

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

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