2 Marks Questions

Question 12 Marks

Diagonals of a quadrilateral are perpendicular to each other. Is such a quadrilateral always a rhombus? Give a figure to justify your answer.

Answer

False, it is not necessary that a quadrilateral having perpendicular diagonals is a rhombus.

e.g., Consider a trapezium ABCD in which AB || CD.

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Question 22 Marks

The ratio between exterior angle and interior angle of a regular polygon is $1 : 5$. Find the number of sides of the polygon.

Answer

Let the exterior angle & interior angle be $x$ and $5x$ , respectively.
The, $x + 5x = 180^\circ $
$\Rightarrow6\text{x}=180^\circ$
$\Rightarrow\text{x}=\frac{180^\circ}{6}=30^\circ$
$\therefore$The number of sides $\Rightarrow\frac{360^\circ}{\text{Exterior angle}}=\frac{360^\circ}{30^\circ}=12^\circ$

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Question 32 Marks

In the figure, find the value of $x$.

Answer

The given figure is a pentagon.
Sum of all the exterior angles of a pentagon is $360^\circ $
$\therefore $ $92^\circ + 20^\circ + 85^\circ + x + 89^\circ = 360^\circ $
$\Rightarrow 286^\circ + x = 360^\circ $
$ \Rightarrow x = 360^\circ - 280^\circ $
$ \Rightarrow x = 74^\circ $

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Question 42 Marks

Both the pairs of opposite angles of a quadrilateral are equal and supplementary. Find the measure of each angle.

Answer

Let $ABCD$ be a quadrilateral, such that $\angle\text{A}=\angle\text{C},\angle\text{B}=\angle\text{D}$ and also $\angle\text{A}+\angle\text{C}=180^\circ\angle\text{B}+\angle\text{D}=180^\circ$
Now, $\angle\text{A}+\angle\text{A}=180^\circ$
$2\angle\text{A}=90^\circ=180^\circ$
$\Rightarrow\angle\text{A}=90^\circ$ Similarly,
$\Rightarrow\angle\text{B}=90^\circ$ Hence, each angle is a right angle.

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Question 52 Marks

Two sticks each of length $7\ cm$ are crossing each other such that they bisect each other at right angles. What shape is formed by joining their end points? Give reason.

Answer

Sticks can be treated as the diagonals of a quadrilateral.
Now, since the diagonals (sticks) are bisecting each other at right angles, therefore the shape formed by joining their end points will be a rhombus.

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Question 62 Marks

A playground in the town is in the form of a kite. The perimeter is $106m$. If one of its sides is $23m$, what are the lengths of other three sides?

Answer

Let the length of other non-consecutive side be $x\ cm$. Then, we have, perimeter of playground
$= 23 + 23 + x + x$
$\Rightarrow 106 = 2 (23 + x)$
$\Rightarrow 46 + 2x = 106$
$2x = 106 - 46$

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Question 72 Marks

In parallelogram $ABCD$, find $\angle\text{B}.\ \angle\text{C}\ \text{and}\angle\text{D}.$

Answer

In a parallelogram, the opposite angles are equal, therefore $\angle\text{C}=\angle\text{A}=80^\circ$ Also, adjacent angles are supplementary.
$\therefore\angle\text{A}+\angle\text{B}=180^\circ$
$\Rightarrow80+\angle\text{B}=180^\circ$
$\Rightarrow\angle\text{B}=180^\circ-80^\circ=100^\circ$
Now, $\angle\text{B}=\angle\text{D}=100^\circ$
$\therefore\angle\text{D}=100^\circ.$

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Question 82 Marks

In rectangle $PAIR$, find $\angle\text{ARI},\angle\text{RMI}\ \text{and}\angle\text{PMA}.$

Answer

Given, $\angle\text{RAI}=35^\circ$
$\angle\text{PRA}=35^\circ$
$\angle\text{ARI}=90^\circ-\angle\text{PRA}=90^\circ-35^\circ=35^\circ$
$\text{AM}=\text{IM},\angle\text{MIA}=\angle\text{MAI}=35^\circ$
$\text{In}\triangle\text{AMI},\angle\text{RMI}=\angle\text{MAI}+\angle\text{MIA}=70^\circ$ [exterior angle] Also, $ \angle\text{RMI}=\angle\text{PMA}$
$ \angle\text{PMA}=70^\circ$ [Vertically opposite angles]

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Question 92 Marks

Find the value of $x$ in the trapezium $ABCD$ given below.

Answer

Given, a trapezium $ABCD$ in which $\angle=(\text{x}-20)^\circ,\angle\text{D}=(\text{x}+40)^\circ$
Since, in a trapezium, the angles on either side of the base are supplementary, therefore
$(\text{x}-20)+(\text{x}+40)=180^\circ$
$\Rightarrow\text{x}-20+40=180^\circ$
$\Rightarrow\text{2x}=20=180^\circ$
$\Rightarrow\text{2x}=(180^\circ+20^\circ)=160^\circ$
$\Rightarrow\text{x}=80^\circ.$

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Question 102 Marks

Two adjacent angles of a parallelogram are in the ratio $1 : 3$. Find its angles.

Answer

Let the adjacent angles of a parallelogram be $x$ and $8c$.
Then, we have $x + (3x) = 180^\circ $ [adjacent angles of parallelogram are supplementary]
$\Rightarrow 4x = 180^\circ $
$ \Rightarrow x = 45^\circ $
Thus, the angles are $45^\circ , 135^\circ .$
Hence, the angles are $45^\circ , 135^\circ , 45^\circ , 135^\circ $. [opposite angles in a parallelogram are equal]

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Question 112 Marks

A playground in the town is in the form of a kite. The perimeter is $106$ metres. If one of its sides is $23$ metres, what are the lengths of other three sides?

Answer

Given, a rectangle $READ$, in which$\angle\text{ROD}=60^\circ$
$\therefore\angle\text{EOA}=180^\circ-60^\circ=120^\circ$
Now, in $\triangle\text{EOA}, \angle\text{OEA}=\angle\text{OAE}=30^\circ$ [$OE$ & $OA$ equals sides make equal angles] $\therefore\angle\text{EAR}=30^\circ,\angle\text{RAD}=90^\circ-\angle\text{EAR}=60^\circ$ and $\angle\text{ARI}=\angle\text{EOA}=120^\circ$

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Question 122 Marks

Find the values of $x$ and $y$ in the following kite.

Answer

The given figure is a kite In a kite, one pair of opposite angles are equal.
$\therefore\text{y}=110^\circ$
Now, by the angles sum property of a quadrilateral,
we have $110^\circ+ 60^\circ+ 110^\circ+ x = 360^\circ$
$\Rightarrow x = 360^\circ− 280^\circ$
$\Rightarrow x = 80^\circ$

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Question 132 Marks

Two angles of a quadrilateral are each of measure $75^\circ$ and the other two angles are equal. What is the measure of these two angles? Name the possible figures so formed.

Answer

Let $ABCD$ be a quadrilateral where $\angle\text{A}+\angle\text{B}+\angle\text{C}=180^\circ\text{and}\angle\text{B}=\angle\text{D}=\text{x}$
Then, by the angle sum property of a quadrilateral $\angle\text{A}+\angle\text{B}+\angle\text{C}=360^\circ$ $\Rightarrow75^\circ+\text{x+75}^\circ+\text{x}=360^\circ$
$\Rightarrow2\text{x}=360^\circ-150^\circ$
$\Rightarrow2\text{x}=210^\circ$
$\Rightarrow2\text{x}=105^\circ$
Thus, other two angles are of $105^\circ$ each.
Since, opposite angles are equal, therefore the quadrilateral is a parallelogram.

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Question 142 Marks

A line $l$ is parallel to line m and a transversal p interesects them at $X, Y$ respectively. Bisectors of interior angles at $X$ and $Y$ interesct at $P$ and $Q$. Is $PXQY$ a rectangle? Given reason.

Answer

Number of sides in pentagon is $5$ and in decagon is $10$.
Now, exterior angle of a regular pentagon $=\frac{360^\circ}{5}=72^\circ$
Exterior angle of a regular decagon $=\frac{360^\circ}{10}=36^\circ$
Required ratio $=\frac{72}{36}=2:1$
So, the ratio between these two angles is $2 : 1$.

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