Understanding Shapes — Multiple Choice Questions

Sum of all interior angles of a 11-sided polygon is

1. 1620°

2. 1440°

3. 1260°

4. none of these

Answer

According to the properties of a polygon, if a polygon has n sides, then the sum of measures of all its interior angles is (n - 2) × 180°.

Here, n = 11.

Sum of all interior angles = (11 - 2) x 180°

= 9 x 180°

= 1620°

Hence, option 1 is the correct option.

If each interior angle of a regular polygon is 144°, then number of sides of polygon is

1. 8

2. 9

3. 10

4. 11

Answer

It is given that each interior angle of a regular polygon is 144°.

As the sum of an interior angle and its adjacent exterior angle is 180°,

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

According to the properties of a polygon, if each exterior angle of a regular polygon is $x\degree$, then the number of sides in the regular polygon is $\dfrac{360}{x}$.

Number of sides = $\dfrac{360\degree}{36\degree}$

= 10

Hence, option 3 is the correct option.

If the sum of all interior angles of a polygon is 1260°, then number of sides of polygon is

1. 6

2. 7

3. 8

4. 9

Answer

It is given that the sum of all interior angles of a polygon is 1260°.

According to the properties of a polygon, if a polygon has n sides, then the sum of measures of all its interior angles is (n - 2) × 180°.

⇒ (n - 2) × 180° = 1260°

⇒ n - 2 = $\dfrac{1260\degree}{180\degree}$

⇒ n - 2 = 7

⇒ n = 7 + 2

⇒ n = 9

Hence, option 4 is the correct option.

The sum of all exterior angles of a pentagon is

1. 590°

2. 360°

3. 180°

4. none of these

Answer

According to the properties of a polygon, the sum of all the exterior angles of a polygon is always 360°.

A pentagon is a polygon of 5 sides, so the sum of all its exterior angles is also 360°.

Hence, option 2 is the correct option.

If the ratio between an exterior and interior angle of a regular polygon is 1 : 5, then the number of sides of regular polygon is

1. 11

2. 12

3. 13

4. 14

Answer

It is given that the ratio between an exterior and interior angle of a regular polygon is 1 : 5.

Let the common factor be a. So, the exterior angle is a and the interior angle is 5a.

As the sum of an interior angle and its adjacent exterior angle is 180°,

⇒ a + 5a = 180°

⇒ 6a = 180°

⇒ a = $\dfrac{180\degree}{6}$

⇒ a = 30°

So, each exterior angle is 30°.

According to the properties of a polygon, if each exterior angle of a regular polygon is $x\degree$, then the number of sides in the regular polygon is $\dfrac{360}{x}$.

Number of sides = $\dfrac{360\degree}{30\degree}$

= 12

Hence, option 2 is the correct option.

In the adjoining figure, the value of x is

1. 140°

2. 50°

3. 130°

4. 40°

Answer

From the figure, the sides of the triangle have been produced in order, so 140°, 90° and x are the exterior angles of the triangle.

According to the properties of a polygon, the sum of all the exterior angles of a polygon is 360°.

⇒ 140° + 90° + x = 360°

⇒ 230° + x = 360°

⇒ x = 360° - 230°

⇒ x = 130°

Hence, option 3 is the correct option.

In the adjoining figure, the value of x is

1. 120°

2. 130°

3. 140°

4. 150°

Answer

From the figure, the side of the quadrilateral has been produced and the exterior angle so formed is 120°.

As the sum of an interior angle and its adjacent exterior angle is 180°,

Interior angle at that vertex = 180° - 120° = 60°

As we know, the sum of all the interior angles of a quadrilateral is 360°.

⇒ 60° + x + 110° + 60° = 360°

⇒ 230° + x = 360°

⇒ x = 360° - 230°

⇒ x = 130°

Hence, option 2 is the correct option.

In the adjoining figure, the value of x + y + z + w is

1. 180°

2. 270°

3. 300°

4. 360°

Answer

From the figure, the sides of the quadrilateral have been produced in order, so x, y, z and w are the exterior angles of the quadrilateral.

According to the properties of a polygon, the sum of all the exterior angles of a polygon is always 360°.

So, x + y + z + w = 360°

Hence, option 4 is the correct option.

In the adjoining figure, the value of x + y is

1. 180°

2. 190°

3. 170°

4. 160°

Answer

From the figure, 110°, y, 60° and x are the interior angles of the quadrilateral.

As we know, the sum of all the interior angles of a quadrilateral is 360°.

⇒ 110° + y + 60° + x = 360°

⇒ x + y + 170° = 360°

⇒ x + y = 360° - 170°

⇒ x + y = 190°

Hence, option 2 is the correct option.

The lengths of two adjacent sides of a parallelogram are in the ratio 1 : 2. If the perimeter of parallelogram is 60 cm, then length of its sides are

1. 6 cm, 12 cm

2. 8 cm, 16 cm

3. 9 cm, 18 cm

4. 10 cm, 20 cm

Answer

It is given that the lengths of two adjacent sides of a parallelogram are in the ratio 1 : 2 and the perimeter of the parallelogram is 60 cm.

Let the common factor be a. So, the adjacent sides are a cm and 2a cm.

As the opposite sides of a parallelogram are of equal length,

Perimeter = a + 2a + a + 2a

⇒ 60 = 6a

⇒ a = $\dfrac{60}{6}$

⇒ a = 10

Thus, the sides are:

a = 10 cm

2a = 2 x 10 = 20 cm

Hence, option 4 is the correct option.

In the adjoining figure ABCD is a parallelogram, the values of x and y respectively are

1. 1 cm, 1 cm

2. 2 cm, 1 cm

3. 1 cm, 2 cm

4. 2 cm, 2 cm

Answer

From the figure, in parallelogram ABCD the diagonals AC and BD intersect at O, where AO = 6 cm, OC = 5x + 1, BO = 4 cm and OD = y + 3.

As the diagonals of a parallelogram bisect each other,

OC = AO

⇒ 5x + 1 = 6

⇒ 5x = 6 - 1

⇒ 5x = 5

⇒ x = $\dfrac{5}{5}$

⇒ x = 1

Also,

OD = BO

⇒ y + 3 = 4

⇒ y = 4 - 3

⇒ y = 1

Hence, option 1 is the correct option.

In the adjoining figure ABCD is a parallelogram, the values of x, y and z respectively are

1. 60°, 60°, 70°

2. 60°, 70°, 60°

3. 70°, 60°, 60°

4. none of these

Answer

From the figure, in parallelogram ABCD the side AB has been produced beyond B and the exterior angle so formed is 70°.

As the sum of an interior angle and its adjacent exterior angle is 180°,

⇒ ∠ABC + 70° = 180°

⇒ ∠ABC = 180° - 70°

⇒ ∠ABC = 110°

From the figure,

⇒ 50° + x = ∠ABC

⇒ 50° + x = 110°

⇒ x = 110° - 50°

⇒ x = 60°

As AD ∥ BC and BD is a transversal,

y = ∠ADB = ∠DBC = x = 60° (alternate angles are equal)

As the opposite angles of a parallelogram are of equal measure,

z = ∠BCD = ∠DAB

As the adjacent angles of a parallelogram are supplementary,

⇒ ∠DAB = 180° - ∠ABC

⇒ ∠DAB = 180° - 110°

⇒ ∠DAB = 70°

So, z = 70°

Hence, option 1 is the correct option.

In the adjoining figure ABCD is a rectangle, the value of angle x is

1. 60°

2. 90°

3. 120°

4. none of these

Answer

From the figure, in rectangle ABCD the diagonals intersect at O and ∠OAB = 30°.

As the diagonals of a rectangle are equal and bisect each other,

OA = OB

So, ΔAOB is an isosceles triangle.

⇒ ∠OBA = ∠OAB = 30° (angles opposite to equal sides are equal)

In ΔAOB, the sum of all the angles of a triangle is 180°.

⇒ ∠OAB + ∠OBA + ∠AOB = 180°

⇒ 30° + 30° + ∠AOB = 180°

⇒ 60° + ∠AOB = 180°

⇒ ∠AOB = 180° - 60°

⇒ ∠AOB = 120°

As x and ∠AOB are vertically opposite angles,

x = ∠AOB = 120°

Hence, option 3 is the correct option.

In a rhombus ABCD, diagonals AC and BD are respectively 8 cm and 6 cm, the length of each side of rhombus is

1. 7 cm

2. 5 cm

3. 6 cm

4. 8 cm

Answer

It is given that in rhombus ABCD, the diagonals AC and BD are respectively 8 cm and 6 cm.

Let the diagonals intersect at O.

As the diagonals of a rhombus bisect each other,

AO = $\dfrac{8}{2}$ = 4 cm

BO = $\dfrac{6}{2}$ = 3 cm

As the diagonals of a rhombus intersect at right angles, ∠AOB = 90°.

So, ΔAOB is a right angled triangle.

By Pythagoras theorem,

⇒ AB² = AO² + BO²

⇒ AB² = 4² + 3²

⇒ AB² = 16 + 9

⇒ AB² = 25

⇒ AB = $\sqrt{25}$

⇒ AB = 5 cm

Hence, option 2 is the correct option.

In the adjoining figure ABCD is a square, the value of angle $x$ is

1. 30°

2. 45°

3. 60°

4. not possible to find

Answer

From the figure, in square ABCD the diagonals AC and BD are drawn and $x$ = ∠ABD.

As we know, the diagonals of a square bisect the angles of the square.

So, the diagonal BD bisects ∠ABC.

$\Rightarrow x = \dfrac{1}{2} \times \angle ABC \\[1em] \Rightarrow x = \dfrac{1}{2} \times 90\degree \\[1em] \Rightarrow x = 45\degree$

Hence, option 2 is the correct option.

In the adjoining figure, ABCD is a kite, the value of angle x is

1. 86°

2. 100°

3. 104°

4. none of these

Answer

From the figure, in kite ABCD, AD = DC and AB = BC, the diagonals AC and BD intersect at O, ∠DCA = 50° and ∠ABD = 36°.

In ΔADC, AD = DC, so ΔADC is an isosceles triangle.

⇒ ∠DAC = ∠DCA = 50° (angles opposite to equal sides are equal)

As the diagonals of a kite intersect at right angles,

∠AOB = 90°

In ΔAOB, the sum of all the angles of a triangle is 180°.

⇒ ∠OAB + ∠ABO + ∠AOB = 180°

⇒ ∠OAB + 36° + 90° = 180°

⇒ ∠OAB + 126° = 180°

⇒ ∠OAB = 180° - 126°

⇒ ∠OAB = 54°

From the figure,

x = ∠DAB = ∠DAC + ∠OAB

⇒ x = 50° + 54°

⇒ x = 104°

Hence, option 3 is the correct option.

In the adjoining figure, ABCD is an isosceles trapezium, the values of x, y and z respectively are

1. 110°, 110°, 70°

2. 110°, 70°, 110°

3. 70°, 110°, 110°

4. none of these.

Answer

From the figure, ABCD is an isosceles trapezium in which AB ∥ DC, AD = BC and ∠A = 70°.

As the angles on the same base of an isosceles trapezium are equal,

z = ∠B = ∠A = 70°

As AB ∥ DC and AD is a transversal, the co-interior angles ∠A and ∠D are supplementary.

⇒ x + ∠A = 180°

⇒ x + 70° = 180°

⇒ x = 180° - 70°

⇒ x = 110°

As the angles on the same base of an isosceles trapezium are equal,

y = ∠C = ∠D = x = 110°

Hence, option 1 is the correct option.