Excercise 3.3 Understanding Quadrilaterals- NCERT Solutions Class 8

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Chapter 3 Ex.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,

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

Chapter 3 Ex.3.3 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,

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

Chapter 3 Ex.3.3 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.

Chapter 3 Ex.3.3 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.

Chapter 3 Ex.3.3 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 \(72^\circ\).

Chapter 3 Ex.3.3 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 \(90^\circ\).

Chapter 3 Ex.3.3 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,

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

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

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

Chapter 3 Ex.3.3 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.

Chapter 3 Ex.3.3 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 \(180^\circ.\)

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

Chapter 3 Ex.3.3 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 \(ML\).

Chapter 3 Ex.3.3 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,

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

Chapter 3 Ex.3.3 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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