Rectilinear Figures — Exercise 13(A)

One angle of a seven-sided polygon is 114° and each of the other six angles is x°. Then the magnitude of x is :

1. 131°

2. 132°

3. 135°

4. 130°

Answer

By Formula,

Sum of interior angles of an 'n' sided polygon = (2n - 4) × 90°.

∴ Sum of interior angles of a seven-sided polygon = (2 × 7 - 4) × 90°

= (14 - 4) × 90°

= 10 × 90° = 900°.

Given,

One angle of a seven-sided polygon is 114° and each of the other six angles is x°.

∴ 114° + 6x = 900°

⇒ 6x = 900° - 114°

⇒ 6x = 786°

⇒ x = $\dfrac{786°}{6}$ = 131°.

Hence, Option 1 is the correct option.

In a parallelogram ABCD, ∠A - ∠C is equal to :

1. 90°

2. 120°

3. 0°

4. 180°

Answer

We know that,

Opposite angles of a parallelogram are equal.

∴ ∠A = ∠C = x (let)

∴ ∠A - ∠C = x - x = 0°.

Hence, Option 3 is the correct option.

If each interior angle of a polygon is 144°; the number of sides in it is :

1. 5

2. 10

3. 6

4. 7

Answer

By formula,

Each interior angle of a regular polygon = $\dfrac{(2n - 4) \times 90°}{n}$

$\therefore \dfrac{(2n - 4) \times 90°}{n} = 144° \\[1em] \Rightarrow 180°.n - 360° = 144°.n \\[1em] \Rightarrow 180°.n - 144°.n = 360° \\[1em] \Rightarrow 36°.n = 360° \\[1em] \Rightarrow n = \dfrac{360°}{36°} = 10.$

Hence, Option 2 is the correct option.

The sum of the interior angles of a regular polygon is equal to six times the sum of its exterior angles. The number of sides of the polygon is :

1. 14

2. 10

3. 12

4. 16

Answer

Let n be the number of sides of the polygon.

By formula,

Sum of interior angles of an 'n' sided regular polygon = (2n - 4) × 90°.

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

Given,

The sum of the interior angles of a regular polygon is equal to six times the sum of its exterior angles.

∴ (2n - 4) × 90° = 6 × 360°

⇒ 2n - 4 = $\dfrac{6 \times 360°}{90°}$

⇒ 2n - 4 = 6 × 4

⇒ 2n - 4 = 24

⇒ 2n = 24 + 4

⇒ 2n = 28

⇒ n = $\dfrac{28}{2}$ = 14.

Hence, Option 1 is the correct option.

An exterior angle and an interior angle of a regular polygon are in the ratio 2 : 7. The number of sides in the polygon is :

1. 12

2. 6

3. 4

4. 9

Answer

Let n be the number of sides of the polygon.

Given,

An exterior angle and an interior angle of a regular polygon are in the ratio 2 : 7.

By formula,

Each interior angle of a regular polygon = $\dfrac{(2n - 4) × 90°}{n}$

Each exterior angle of a regular polygon = $\dfrac{360°}{n}$

$\Rightarrow \dfrac{\dfrac{360°}{n}}{\dfrac{(2n - 4) × 90°}{n}} = \dfrac{2}{7} \\[1em] \Rightarrow \dfrac{360° \times n}{(2n - 4) \times 90° \times n} = \dfrac{2}{7} \\[1em] \Rightarrow \dfrac{4}{2n - 4} = \dfrac{2}{7} \\[1em] \Rightarrow 2(2n - 4) = 28 \\[1em] \Rightarrow 4n - 8 = 28 \\[1em] \Rightarrow 4n = 28 + 8 \\[1em] \Rightarrow 4n = 36 \\[1em] \Rightarrow n = \dfrac{36}{4} = 9.$

Hence, Option 4 is the correct option.

The sum of the interior angles of a polygon is four times the sum of its exterior angles. Find the number of sides in the polygon.

Answer

Let n be the number of sides of the polygon.

By formula,

Sum of interior angles of an 'n' sided polygon = (2n - 4) × 90°.

Sum of exterior angles of a polygon = 360°.

Given,

The sum of the interior angles of a polygon is four times the sum of its exterior angles.

⇒ (2n - 4) × 90° = 4 × 360°

⇒ (2n - 4) = $\dfrac{4 \times 360°}{90°}$

⇒ (2n - 4) = 4 × 4

⇒ 2n - 4 = 16

⇒ 2n = 16 + 4

⇒ 2n = 20

⇒ n = $\dfrac{20}{2}$ = 10.

Hence, number of sides in polygon = 10.

The angles of a pentagon are in the ratio 4 : 8 : 6 : 4 : 5. Find each angle of the pentagon.

Answer

By formula,

Sum of interior angles of an 'n' sided polygon = (2n - 4) × 90°.

Sum of interior angles of a pentagon = [2 × 5 - 4] × 90°

= [10 - 4] × 90°

= 6 × 90°

= 540°.

Given,

The angles of a pentagon are in the ratio 4 : 8 : 6 : 4 : 5.

Let angles be 4x, 8x, 6x, 4x and 5x.

⇒ 4x + 8x + 6x + 4x + 5x = 540°

⇒ 27x = 540°

⇒ x = $\dfrac{540°}{27}$ = 20°.

⇒ 4x = 4(20°) = 80°, 8x = 8(20°) = 160°, 6x = 6(20°) = 120°, 4x = 4(20°) = 80° and 5x = 5(20°) = 100°.

Hence, angles of pentagon are 80°, 160°, 120°, 80° and 100°.

One angle of a six-sided polygon is 140° and the other angles are equal. Find the measure of each equal angle.

Answer

By formula,

Sum of interior angles of an 'n' sided polygon = (2n - 4) × 90°.

Sum of interior angles of a six-sided polygon = [2 × 6 - 4] × 90°

= [12 - 4] × 90°

= 8 × 90°

= 720°.

Given,

One angle of a six-sided polygon is 140° and the other angles are equal.

∴ 140° + 5x = 720°

⇒ 5x = 720° - 140°

⇒ 5x = 580°

⇒ x = $\dfrac{580°}{5}$ = 116°.

Hence, each equal angle of a six-sided polygon = 116°.

In a polygon, there are 5 right angles and the remaining angles are equal to 195° each. Find the number of sides in the polygon.

Answer

Let n be the number of sides of the polygon.

By formula,

Sum of interior angles of an 'n' sided polygon = (2n - 4) × 90°.

Given,

In the polygon, there are 5 right angles and the remaining angles are equal to 195° each.

∴ 5 × 90° + (n - 5) × 195° = (2n - 4) × 90°

⇒ 450° + 195°.n - 975° = 180°.n - 360°

⇒ 195°.n - 180°.n = 975° - 450° - 360°

⇒ 15°.n = 165°

⇒ n = $\dfrac{165°}{15°}$ = 11.

Hence, no. of sides in the polygon = 11.

Three angles of a seven sided polygon are 132° each and remaining four angles are equal. Find the value of each equal angle.

Answer

By formula,

Sum of interior angles of an 'n' sided polygon = (2n - 4) × 90°.

Sum of interior angles of 7 sided polygon = [2 × 7 - 4] × 90°

= [14 - 4] × 90°

= 10 × 90°

= 900°.

Given,

Three angles of a seven sided polygon are 132° each and remaining four angles are equal. Let each equal angle be x.

⇒ 3 × 132° + 4x = 900°

⇒ 396° + 4x = 900°

⇒ 4x = 900° - 396°

⇒ 4x = 504°

⇒ x = $\dfrac{504°}{4}$ = 126°.

Hence, each equal angle = 126°.

Two angles of an eight sided polygon are 142° and 176°. If the remaining angles are equal to each other; find the magnitude of each of the equal angles.

Answer

By formula,

Sum of interior angles of an 'n' sided polygon = (2n - 4) × 90°.

Sum of interior angles of 8 sided polygon = [2 × 8 - 4] × 90°

= [16 - 4] × 90°

= 12 × 90°

= 1080°.

Given,

Two angles of an eight sided polygon are 142° and 176° and remaining angles are equal. Let each equal angle be x.

⇒ 142° + 176° + 6x = 1080°

⇒ 318° + 6x = 1080°

⇒ 6x = 1080° - 318°

⇒ 6x = 762°

⇒ x = $\dfrac{762°}{6}$ = 127°.

Hence, each equal angle = 127°.

In a pentagon ABCDE, AB is parallel to DC and ∠A : ∠E : ∠D = 3 : 4 : 5. Find angle E.

Answer

By formula,

Sum of interior angles of an 'n' sided polygon = (2n - 4) × 90°.

Sum of interior angles of 5 sided polygon = [2 × 5 - 4] × 90°

= [10 - 4] × 90°

= 6 × 90°

= 540°.

We know that,

Sum of interior angles on the same side of transversal are supplementary.

∴ ∠B + ∠C = 180°.

Given,

∠A : ∠E : ∠D = 3 : 4 : 5

Let ∠A = 3x, ∠E = 4x and ∠D = 5x.

∴ ∠A + ∠B + ∠C + ∠D + ∠E = 540°

⇒ 3x + 180° + 5x + 4x = 540°

⇒ 12x = 540° - 180°

⇒ 12x = 360°

⇒ x = $\dfrac{360°}{12}$ = 30°.

⇒ ∠E = 4x = 4(30°) = 120°.

Hence, ∠E = 120°.

AB, BC and CD are the three consecutive sides of a regular polygon. If ∠BAC = 15°; find,

(i) each interior angle of the polygon.

(ii) each exterior angle of the polygon.

(iii) number of sides of the polygon.

Answer

(i) In △ ABC,

⇒ AB = BC (As, ABCD is a regular polygon)

⇒ ∠BCA = ∠BAC = 15° (In a triangle angles opposite to equal sides are equal)

By angle sum property of triangle,

⇒ ∠BCA + ∠BAC + ∠ABC = 180°

⇒ 15° + 15° + ∠ABC = 180°

⇒ 30° + ∠ABC = 180°

⇒ ∠ABC = 180° - 30° = 150°.

Since, each interior angle of a regular polygon are equal.

Hence, each interior angle of a regular polygon = 150°.

(ii) We know that,

At each vertex of every polygon,

⇒ Exterior angle + Interior angle = 180°

⇒ Exterior angle + 150° = 180°

⇒ Exterior angle = 180° - 150° = 30°.

Hence, each exterior angle of a regular polygon = 30°.

(iii) Let n be the number of sides in the polygon.

By formula,

Sum of interior angles of an 'n' sided polygon = (2n - 4) × 90°.

∴ 150°.n = (2n - 4) × 90°

⇒ 150°.n = 180°.n - 360°

⇒ 180°.n - 150°.n = 360°

⇒ 30°.n = 360°

⇒ n = $\dfrac{360°}{30°}$ = 12.

Hence, no. of sides in polygon = 12.

The ratio between an exterior angle and an interior angle of a regular polygon is 2 : 3. Find the number of sides in the polygon.

Answer

Given,

Ratio between an exterior angle and an interior angle of a regular polygon is 2 : 3.

Let exterior angle be 2x and interior angle be 3x.

We know that,

At each vertex of every polygon,

⇒ Exterior angle + Interior angle = 180°

⇒ 2x + 3x = 180°

⇒ 5x = 180°

⇒ x = $\dfrac{180°}{5}$ = 36°.

⇒ 2x = 2 × 36° = 72°, 3x = 3 × 36° = 108°.

By formula,

If each exterior angle of a regular polygon is x°, the number of sides in it = $\dfrac{360°}{x°}$

No. of sides in a regular polygon with exterior angle = 72° is $\dfrac{360°}{72°}$ = 5.

Hence, no. of sides in polygon = 5.