1 Marks Question

Question 11 Mark

Find the measure of exterior angle of a regular pentagon and an exterior angle of a regular decagon. What is the ratio between these two angles?

Answer

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

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Question 21 Mark

Each interior angle of a polygon is $108^{\circ}.$ Find the number of sides of the polygon.

Answer

Given, interior angle $=108^{\circ}$
$\therefore$ Exterior angle $=180^{\circ}-108^{\circ}=72^{\circ}$
$\begin{aligned} \therefore \text { Number of sides } & =\frac{360^{\circ}}{\text { Exterior angle }} \\ & =\frac{360^{\circ}}{72^{\circ}}=5\end{aligned}$

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Question 31 Mark

In the following figure, find the value of x.

Answer

We know that sum of all the exterior angles of a pentagon is $360^{\circ}$.
$\therefore 92^{\circ}+20^{\circ}+85^{\circ}+x^{\circ}+89^{\circ}=360^{\circ}$
$\Rightarrow \quad 286^{\circ}+x^{\circ}=360^{\circ}$
$\Rightarrow \quad x^{\circ}=360^{\circ}-286^{\circ}=74^{\circ}$

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Question 41 Mark

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

Answer

In the trapezium $A B C D$, we have
$A B \| C D$
Also, sum of interior angles $B$ and $C$ is $180^{\circ}$.
$\therefore(x+20)^{\circ}+(x-30)^{\circ}=180^{\circ}$
$\Rightarrow \quad 2 x^{\circ}-10^{\circ}=180^{\circ}$
$\Rightarrow \quad 2 x^{\circ}=190^{\circ}$
$\Rightarrow \quad x^{\circ}=95^{\circ}$

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Question 51 Mark

If four angles of a quadrilateral are in the ratio 3: 8: 10: 3, then find its all angles.

Answer

Let the angles be $3 x, 8 x, 10 x$ and $3 x$.
So, $\quad 3 x+8 x+10 x+3 x=360^{\circ}\quad$ [angle sum property of a quadrilateral]
$\Rightarrow \quad 24 x=360^{\circ}$
$\Rightarrow \quad x=360^{\circ} \times \frac{1}{24}=15^{\circ}$
Thus, the angles are $3 \times 15^{\circ}=45^{\circ}, 8 \times 15^{\circ}=120^{\circ}$, $10 \times 15^{\circ}=150^{\circ}$ and $3 \times 15^{\circ}=45^{\circ}$.

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Question 61 Mark

Find an exterior angle of a regular octagon and an exterior angle of a regular heptagon. What is the difference between these two angles?

Answer

Exterior angle of regular octagon $=\frac{360^{\circ}}{8}=45^{\circ}$
Exterior angle of regular heptagon $=\frac{360^{\circ}}{7}=51.42^{\circ}$
So, difference between these two angle is
$=51.42^{\circ}-45^{\circ}=6.42^{\circ}$

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Question 71 Mark

Find the interior angle of decagon.

Answer

$\begin{aligned} \text { Exterior angle of decagon } & =\frac{360^{\circ}}{\text { Number of sides }} \\ & =\frac{360^{\circ}}{10}=36^{\circ}\end{aligned}$
$\begin{aligned} \therefore \text { Interior angle } & =180^{\circ}-\text { Exterior angle } \\ & =180^{\circ}-36^{\circ}=144^{\circ}\end{aligned}$

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Question 81 Mark

In the following figure, find the value of x + y + z + w.

Answer

By linear pair property and angles sum property of a quadrilateral,
$\left(180^{\circ}-x\right)+\left(180^{\circ}-y\right)+\left(180^{\circ}-z\right)+\left(180^{\circ}-w\right)=360^{\circ}$
$\Rightarrow \quad 720^{\circ}-(x+y+z+w)=360^{\circ}$
$\Rightarrow \quad(x+y+z+w)=720^{\circ}-360^{\circ}=360^{\circ}$

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Question 91 Mark

In the following figure, find the value of x + y + z.

Answer

By linear pair property and angles sum property of a triangle,
$\left(180^{\circ}-x\right)+\left(180^{\circ}-y\right)+\left(180^{\circ}-z\right)=180^{\circ}$
$\Rightarrow \quad 540^{\circ}-x-y-z=180^{\circ}$
$\Rightarrow \quad x+y+z=540^{\circ}-180^{\circ}=360^{\circ}$

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