Polygons

A closed figure which does not intersect itself is called a simple closed figure.

(i) The figure begins and ends at the same point and does not intersect itself.

Hence, the given figure is a simple closed figure.

(ii) The figure begins and ends at the same point and does not intersect itself.

Hence, the given figure is a simple closed figure.

(iii) The figure intersects itself.

Hence, the given figure is not a simple closed figure.

(iv) The figure begins and ends at the same point and does not intersect itself.

Hence, the given figure is a simple closed figure.

(v) The figure intersects itself.

Hence, the given figure is not a simple closed figure.

(vi) The figure intersects itself.

Hence, the given figure is not a simple closed figure.

A simple closed figure bounded by three or more line segments is called a polygon.

(i) The figure is bounded by line segments and a curved line.

Hence, the given figure is not a polygon.

(ii) The figure is a simple closed figure bounded by 4 line segments and one of its angles is more than 180°.

Hence, the given figure is a polygon, namely a concave quadrilateral.

(iii) The figure is a simple closed figure bounded by 6 line segments and at least one of its angles is more than 180°.

Hence, the given figure is a polygon, namely a concave hexagon.

(iv) The figure is a simple closed figure bounded by 6 line segments, having all sides equal and all angles equal.

Hence, the given figure is a polygon, namely a regular hexagon.

(v) The figure is a simple closed figure bounded by 10 line segments and at least one of its angles is more than 180°.

Hence, the given figure is a polygon, namely a concave decagon.

(vi) The figure is a simple closed figure bounded by 12 line segments and at least one of its angles is more than 180°.

Hence, the given figure is a polygon, namely a concave 12-gon.

We know that the sum of the interior angles of a polygon of $n$ sides = $(2n - 4)$ right angles.

A nonagon has 9 sides, so $n = 9$.

Sum of interior angles of a nonagon = (2 × 9 - 4) right angles

= 14 right angles

= 14 × 90°

= 1260°

Hence, the sum of the interior angles of a nonagon is 1260°.

We know that the sum of the interior angles of a polygon of $n$ sides = $(2n - 4)$ right angles.

Here, $n = 16$.

Sum of interior angles of a 16-sided polygon = (2 × 16 - 4) right angles

= 28 right angles

= 28 × 90°

= 2520°

Hence, the sum of the interior angles of a 16-sided polygon is 2520°.

We know that the sum of the interior angles of a polygon of $n$ sides = $(2n - 4)$ right angles.

Since the interior angles of a regular polygon are of the same measure, each interior angle is obtained by dividing the sum of the interior angles by the number of sides.

A decagon has 10 sides, so $n = 10$.

Sum of interior angles of a decagon = (2 × 10 - 4) right angles

= 16 × 90°

= 1440°

Each interior angle of a regular decagon = $\frac{1440^{\circ}}{10}$

= 144°

Hence, each interior angle of a regular decagon is 144°.

We know that the sum of the interior angles of a polygon of $n$ sides = $(2n - 4)$ right angles.

Since the interior angles of a regular polygon are of the same measure, each interior angle is obtained by dividing the sum of the interior angles by the number of sides.

Here, $n = 18$.

Sum of interior angles of an 18-sided polygon = (2 × 18 - 4) right angles

= 32 × 90°

= 2880°

Each interior angle of a regular 18-sided polygon = $\frac{2880^{\circ}}{18}$

= 160°

Hence, each interior angle of a regular 18-sided polygon is 160°.

It is given that five of the angles of a hexagon are each 115°.

We know that the sum of the interior angles of a polygon of $n$ sides = $(2n - 4)$ right angles.

A hexagon has 6 sides, so $n = 6$.

Sum of interior angles of a hexagon = (2 × 6 - 4) right angles

= 8 × 90°

= 720°

Sum of the five given angles = 5 × 115° = 575°

So,

Sixth angle = 720° - 575°

= 145°

Hence, the measure of the sixth angle is 145°.

It is given that the angles of a heptagon are $(x + 3)^{\circ}, (2x + 5)^{\circ}, (x + 8)^{\circ}, (3x + 1)^{\circ}, (5x - 6)^{\circ}, (2x + 9)^{\circ}$ and $(x - 5)^{\circ}$.

We know that the sum of the interior angles of a polygon of $n$ sides = $(2n - 4)$ right angles.

A heptagon has 7 sides, so $n = 7$.

Sum of interior angles of a heptagon = (2 × 7 - 4) right angles

= 10 × 90°

= 900°

So,

$$\begin{aligned}\Rightarrow (x + 3)^{\circ} + (2x + 5)^{\circ} + (x + 8)^{\circ} + (3x + 1)^{\circ} + (5x - 6)^{\circ} + (2x + 9)^{\circ} + (x - 5)^{\circ} = 900^{\circ} \\[1em] \Rightarrow x^{\circ} + 3^{\circ} + 2x^{\circ} + 5^{\circ} + x^{\circ} + 8^{\circ} + 3x^{\circ} + 1^{\circ} + 5x^{\circ} - 6^{\circ} + 2x^{\circ} + 9^{\circ} + x^{\circ} - 5^{\circ} = 900^{\circ} \\[1em] \Rightarrow 15x^{\circ} + 15^{\circ} = 900^{\circ} \\[1em] \Rightarrow 15x^{\circ} = 900^{\circ} - 15^{\circ} \\[1em] \Rightarrow 15x^{\circ} = 885^{\circ} \\[1em] \Rightarrow x^{\circ} = \frac{885^{\circ}}{15} \\[1em] \Rightarrow x = 59\end{aligned}$$

Hence, the value of $\bm{x = 59}$.

It is given that an octagon has three equal angles each of measure 115° and all the remaining angles have equal measure.

We know that the sum of the interior angles of a polygon of $n$ sides = $(2n - 4)$ right angles.

An octagon has 8 sides, so $n = 8$.

Sum of interior angles of an octagon = (2 × 8 - 4) right angles

= 12 × 90°

= 1080°

Sum of the three equal angles = 3 × 115° = 345°

Number of remaining angles = 8 - 3 = 5

Let the measure of each of the remaining angles be $a$.

So,

$$\begin{aligned}\Rightarrow 345^{\circ} + 5a = 1080^{\circ} \\[1em] \Rightarrow 5a = 1080^{\circ} - 345^{\circ} \\[1em] \Rightarrow 5a = 735^{\circ} \\[1em] \Rightarrow a = \frac{735^{\circ}}{5} \\[1em] \Rightarrow a = 147^{\circ}\end{aligned}$$

Hence, the measure of each of the remaining angles is 147°.

It is given that the sum of the interior angles of a polygon is 2160°.

Let the number of sides of the polygon be $n$.

Then, sum of interior angles of the polygon = $(2n - 4)$ right angles = $(2n - 4)$ × 90°.

So,

$$\begin{aligned}\Rightarrow (2n - 4) \times 90^{\circ} = 2160^{\circ} \\[1em] \Rightarrow 2n - 4 = \frac{2160^{\circ}}{90^{\circ}} \\[1em] \Rightarrow 2n - 4 = 24 \\[1em] \Rightarrow 2n = 24 + 4 \\[1em] \Rightarrow 2n = 28 \\[1em] \Rightarrow n = \frac{28}{2} \\[1em] \Rightarrow n = 14\end{aligned}$$

Hence, the polygon has 14 sides.

Let the number of sides of the regular polygon be $n$.

Then, sum of its interior angles = $(2n - 4)$ right angles = $(2n - 4)$ × 90°.

Measure of each interior angle = $\frac{(2n - 4) \times 90^{\circ}}{n}$

$$\begin{aligned}\Rightarrow \frac{(2n - 4) \times 90}{n} = 140 \\[1em] \Rightarrow (2n - 4) \times 90 = 140n \\[1em] \Rightarrow 180n - 360 = 140n \\[1em] \Rightarrow 180n - 140n = 360 \\[1em] \Rightarrow 40n = 360 \\[1em] \Rightarrow n = \frac{360}{40} \\[1em] \Rightarrow n = 9\end{aligned}$$

Hence, the number of sides is 9.

Let the number of sides of the regular polygon be $n$.

Then, sum of its interior angles = $(2n - 4)$ right angles = $(2n - 4)$ × 90°.

Measure of each interior angle = $\frac{(2n - 4) \times 90^{\circ}}{n}$

$$\begin{aligned}\Rightarrow \frac{(2n - 4) \times 90}{n} = 162 \\[1em] \Rightarrow (2n - 4) \times 90 = 162n \\[1em] \Rightarrow 180n - 360 = 162n \\[1em] \Rightarrow 180n - 162n = 360 \\[1em] \Rightarrow 18n = 360 \\[1em] \Rightarrow n = \frac{360}{18} \\[1em] \Rightarrow n = 20\end{aligned}$$

Hence, the number of sides is 20.

Let the number of sides of the regular polygon be $n$.

Then, sum of its interior angles = $(2n - 4)$ right angles = $(2n - 4)$ × 90°.

Measure of each interior angle = $\frac{(2n - 4) \times 90^{\circ}}{n}$

$$\begin{aligned}\Rightarrow \frac{(2n - 4) \times 90}{n} = 150 \\[1em] \Rightarrow (2n - 4) \times 90 = 150n \\[1em] \Rightarrow 180n - 360 = 150n \\[1em] \Rightarrow 180n - 150n = 360 \\[1em] \Rightarrow 30n = 360 \\[1em] \Rightarrow n = \frac{360}{30} \\[1em] \Rightarrow n = 12\end{aligned}$$

Hence, the number of sides is 12.

A decagon has 10 sides, so $n = 10$.

$$\begin{aligned}\text{Each interior angle} = \frac{(2 \times 10 - 4) \times 90^{\circ}}{10} \\[1em] = \frac{16 \times 90^{\circ}}{10} \\[1em] = \frac{1440^{\circ}}{10} \\[1em] = 144^{\circ}\end{aligned}$$

As an exterior angle and the adjacent interior angle together form a linear pair,

Each exterior angle = 180° - Each interior angle

= 180° - 144°

= 36°

Hence, each exterior angle of a regular decagon is 36°.

From the figure, the curve begins and ends at the same point, so it is a closed curve.

However, the curve intersects itself at a point, so it is not a simple closed curve.

Hence, option 2 is the correct option.

The region of a curve is made up of the interior of the curve together with its boundary.

Hence, option 4 is the correct option.

A simple closed figure bounded by three or more line segments is called a polygon.

In option 1, the figure intersects itself, so it is not a simple closed figure. Hence, it is not a polygon.

In option 2, one side of the figure is a curved line, so it is not a polygon.

In option 3, the figure is a simple closed figure bounded by 5 line segments, so it is a polygon.

In option 4, the top of the figure is a curved line, so it is not a polygon.

Hence, option 3 is the correct option.

It is given that each interior angle of a regular polygon is 135°.

Let the number of sides of the polygon be $n$.

Then, sum of its interior angles = $(2n - 4)$ right angles = $(2n - 4)$ × 90°.

$$\begin{aligned}\Rightarrow \frac{(2n - 4) \times 90}{n} = 135 \\[1em] \Rightarrow (2n - 4) \times 90 = 135n \\[1em] \Rightarrow 180n - 360 = 135n \\[1em] \Rightarrow 180n - 135n = 360 \\[1em] \Rightarrow 45n = 360 \\[1em] \Rightarrow n = \frac{360}{45} \\[1em] \Rightarrow n = 8\end{aligned}$$

So, the polygon has 8 sides.

Hence, option 3 is the correct option.

It is given that each exterior angle of a regular polygon is 36°.

As an exterior angle and the adjacent interior angle together form a linear pair,

Each interior angle = 180° - 36° = 144°

Let the number of sides of the polygon be $n$.

Then, sum of its interior angles = $(2n - 4)$ right angles = $(2n - 4)$ × 90°.

$$\begin{aligned}\Rightarrow \frac{(2n - 4) \times 90}{n} = 144 \\[1em] \Rightarrow (2n - 4) \times 90 = 144n \\[1em] \Rightarrow 180n - 360 = 144n \\[1em] \Rightarrow 180n - 144n = 360 \\[1em] \Rightarrow 36n = 360 \\[1em] \Rightarrow n = \frac{360}{36} \\[1em] \Rightarrow n = 10\end{aligned}$$

So, the polygon has 10 sides.

Hence, option 2 is the correct option.

We know that the sum of the exterior angles of a polygon = 360°, so each exterior angle of a regular polygon of $n$ sides = $\frac{360^{\circ}}{n}$.

A decagon has 10 sides.

Each exterior angle = $\frac{360^{\circ}}{10}$

= 36°

As an exterior angle and the adjacent interior angle together form a linear pair,

Each interior angle = 180° - 36°

= 144°

Hence, option 3 is the correct option.

We know that the sum of the interior angles of a polygon of $n$ sides = $(2n - 4)$ right angles.

Here, $n = 18$.

Sum of interior angles of an 18-sided polygon = (2 × 18 - 4) right angles

= 32 × 90°

= 2880°

Since the interior angles of a regular polygon are of the same measure,

Each interior angle = $\frac{2880^{\circ}}{18}$

= 160°

Hence, option 2 is the correct option.

It is given that the angles of a heptagon are $(x - 5)^{\circ}, (x + 7)^{\circ}, (5x - 6)^{\circ}, (2x + 1)^{\circ}, (x + 8)^{\circ}, (2x + 9)^{\circ}$ and $(3x + 1)^{\circ}$.

We know that the sum of the interior angles of a polygon of $n$ sides = $(2n - 4)$ right angles.

A heptagon has 7 sides, so $n = 7$.

Sum of interior angles of a heptagon = (2 × 7 - 4) right angles

= 10 × 90°

= 900°

So,

$$\begin{aligned}\Rightarrow (x - 5)^{\circ} + (x + 7)^{\circ} + (5x - 6)^{\circ} + (2x + 1)^{\circ} + (x + 8)^{\circ} + (2x + 9)^{\circ} + (3x + 1)^{\circ} = 900^{\circ} \\[1em] \Rightarrow x^{\circ} - 5^{\circ} + x^{\circ} + 7^{\circ} + 5x^{\circ} - 6^{\circ} + 2x^{\circ} + 1^{\circ} + x^{\circ} + 8^{\circ} + 2x^{\circ} + 9^{\circ} + 3x^{\circ} + 1^{\circ} = 900^{\circ} \\[1em] \Rightarrow 15x^{\circ} + 15^{\circ} = 900^{\circ} \\[1em] \Rightarrow 15x^{\circ} = 900^{\circ} - 15^{\circ} \\[1em] \Rightarrow 15x^{\circ} = 885^{\circ} \\[1em] \Rightarrow x^{\circ} = \frac{885^{\circ}}{15} \\[1em] \Rightarrow x = 59\end{aligned}$$

Hence, option 1 is the correct option.

We know that the sum of the interior angles of a polygon of $n$ sides = $(2n - 4)$ right angles.

A hexagon has 6 sides, so $n = 6$.

Sum of the angles = {(2 × 6 - 4) × 90}°

= (8 × 90)°

= 720°

Sum of the five given angles = 5 × 115° = 575°

So,

6th angle = 720° - 575°

= 145°

Hence, option 2 is the correct option.

It is given that each interior angle of a regular polygon is 150°.

Let the number of sides of the polygon be $n$.

Then, sum of its interior angles = $(2n - 4)$ right angles = $(2n - 4)$ × 90°.

$$\begin{aligned}\Rightarrow \frac{(2n - 4) \times 90}{n} = 150 \\[1em] \Rightarrow (2n - 4) \times 90 = 150n \\[1em] \Rightarrow 180n - 360 = 150n \\[1em] \Rightarrow 180n - 150n = 360 \\[1em] \Rightarrow 30n = 360 \\[1em] \Rightarrow n = \frac{360}{30} \\[1em] \Rightarrow n = 12\end{aligned}$$

So, the polygon has 12 sides.

Hence, option 3 is the correct option.

(i) A regular polygon of 8 sides is called an octagon.

(ii) Sum of interior angles of a polygon of $n$ sides is $\bm{(2n - 4)}$ right angles.

(iii) A polygon of $n$ sides can be divided into $\bm{(n - 2)}$ triangles.

(iv) Sum of exterior angles of a regular polygon of $n$ sides is 360°.

(v) Each angle of a regular polygon of 12 sides is 150°.

Each angle of a regular polygon = $\frac{(2n - 4) \times 90^{\circ}}{n}$

= $\frac{(2 \times 12 - 4) \times 90^{\circ}}{12}$

= $\frac{20 \times 90^{\circ}}{12}$

= 150°

(i) Each interior angle of a regular hexagon is 60°.

False

Each interior angle of a regular hexagon = $\frac{(2 \times 6 - 4) \times 90^{\circ}}{6}$ = 120°.

(ii) Each exterior angle of a regular pentagon is 72°.

True

Sum of the exterior angles of a polygon = 360°.

Each exterior angle of a regular pentagon = $\frac{360^{\circ}}{5}$ = 72°.

(iii) Sum of interior angles of a nonagon is 1260°.

True

Sum of interior angles of a nonagon = $(2n - 4)$ right angles = (2 × 9 - 4) × 90° = 14 × 90° = 1260°.

(iv) Each interior angle of a regular decagon is 144°.

True

Each interior angle of a regular decagon = $\frac{(2 \times 10 - 4) \times 90^{\circ}}{10}$

= $\frac{16 \times 90^{\circ}}{10}$ = 144°.

(v) Sum of the angles of a pentagon is 6 right angles.

True

Sum of the angles of a pentagon = $(2n - 4)$ right angles = (2 × 5 - 4) right angles = 6 right angles.

A square has all its 4 sides equal and all its 4 angles equal, each being 90°.

So, a square is a regular polygon of 4 sides.

So, Assertion (A) is true.

A polygon having all sides equal and all angles equal is called a regular polygon.

So, Reason (R) is true.

As a square has all its sides equal and all its angles equal, it is a regular polygon. So, Reason (R) is the correct explanation of Assertion (A).

Hence, option 1 is the correct option.

On producing a side of a polygon, an exterior angle is formed and the exterior angle and the adjacent interior angle together form a linear pair.

So, Assertion (A) is true.

For a polygon of $n$ sides, the sum of the interior angles = $(2n - 4)$ right angles.

So, Reason (R) is true.

However, Reason (R) gives the sum of the interior angles of a polygon and does not explain why an exterior angle and the adjacent interior angle form a linear pair. So, Reason (R) is not the correct explanation of Assertion (A).

Hence, option 2 is the correct option.

A curve which does not intersect itself is called a simple curve.

From the figure, the 1st figure (curved square), the 4th figure (circle) and the 5th figure (wavy curve) do not intersect themselves, so they are simple curves.

The 2nd figure (figure of eight), the 3rd figure (crossed figure) and the 6th figure (square with loops) intersect themselves, so they are not simple curves.

So, 3 of the given figures are simple curves.

Hence, option 2 is the correct option.

From the figure, ABCD is a polygon with sides AB, BC, CD and DA.

The two end points of a side of a polygon are called adjacent vertices.

So, the pairs of adjacent vertices are (A, B), (B, C), (C, D) and (D, A).

So, there are 4 pairs of adjacent vertices.

Hence, option 3 is the correct option.

Let the five sided polygon be pentagon ABCDE.

A line segment joining the non-adjacent vertices of a polygon is called its diagonal.

So, the diagonals of pentagon ABCDE are AC, AD, BD, BE and CE.

So, there are 5 diagonals in a five sided polygon.

Hence, option 1 is the correct option.

From the figure, ABCDE is a pentagon with sides AB, BC, CD, DE and EA.

If two sides of a polygon have a common end point, then these sides are called the adjacent sides of the polygon.

So, the pairs of adjacent sides are (AB, BC), (BC, CD), (CD, DE), (DE, EA) and (EA, AB).

So, the number of pairs of adjacent sides is 5, not 3. Thus, statement (1) is incorrect.

The vertices of the polygon are A, B, C, D and E.

So, the number of vertices is 5. Thus, statement (2) is correct.

Hence, option 1 is the correct option.

From the figure, the given polygon has 6 sides. Let the hexagon be ABCDEF.

A line segment joining the non-adjacent vertices of a polygon is called its diagonal.

So, the diagonals of hexagon ABCDEF are AC, AD, AE, BD, BE, BF, CE, CF and DF.

So, the number of diagonals in the given polygon is 9.

Hence, option 2 is the correct option.