Quadrilaterals — Multiple Choice Questions

Three angles of a quadrilateral measure 56°, 115° and 84°. Measure of the fourth angle is :

1. 100°

2. 105°

3. 95°

4. 110°

Answer

Let the fourth angle be x.

Given the three angles are 56°, 115° and 84°:

The sum of the angles of a quadrilateral is 360°

56°+ 115° + 84° + x = 360°

255° + x = 360°

x = 360° - 255°

x = 105°

Hence, option 2 is the correct option.

The angles of a quadrilateral are in the ratio 2 : 4 : 5 : 7. The angles of given quadrilateral are :

1. 40°, 60°, 100°, 140°

2. 40°, 80°, 120°, 120°

3. 40°, 80°, 100°, 140°

4. 40°, 60°, 100°, 160°

Answer

We know that,

The sum of the angles of a quadrilateral is 360°

Given the ratio of the angles is 2 : 4 : 5 : 7.

Let the angles be 2x, 4x, 5x, and 7x.

2x + 4x + 5x + 7x = 360°

18x = 360°

x = 20°

2x = 40°

4x = 80°

5x = 100°

7x = 140°

Hence, option 3 is the correct option.

ABCD is a parallelogram in which ∠A = 72°. Measures of ∠B, ∠C and ∠D respectively will be :

1. 72°, 108°, 108°

2. 108°, 108°, 72°

3. 108°, 72°, 108°

4. none of these

Answer

In a parallelogram, consecutive angles are supplementary and opposite angles are equal.

∠A + ∠B = 180°

72° + ∠B = 180°

∠B = 180° - 72°

∠B = 108°

∠A = ∠C = 72° (Opposite angles of parallelogram are equal.)

∠D = ∠B = 108° (Opposite angles of parallelogram are equal.)

Hence, option 3 is the correct option.

In parallelogram ABCD, if ∠A = 2x + 25° and ∠B = 3x − 5°, then value of x will be :

1. x = 13°

2. x = 23°

3. x = 33°

4. x = 32°

Answer

In a parallelogram, consecutive angles are supplementary and opposite angles are equal.

∠A = 2x + 25° and ∠B = 3x − 5°

∠A + ∠B = 180°

2x + 25° + 3x − 5° = 180°

5x + 20° = 180°

5x = 180° - 20°

5x = 160°

x = 32°.

Hence, option 4 is the correct option.

If one angle of a parallelogram is 30° less than twice the smallest angle, then measure of each angle will be :

1. 60°, 80°, 80°, 140°

2. 70°, 110°, 70°, 110°

3. 60°, 120°, 60°, 120°

4. 75°, 105°, 75°, 105°

Answer

Let the smallest angle of the parallelogram be denoted as x.

Given,

One angle of a parallelogram is 30° less than twice the smallest angle = 2x - 30°.

In a parallelogram, consecutive angles are supplementary and opposite angles are equal.

x + 2x - 30° = 180°

3x - 30° = 180°

3x = 180° + 30°

3x = 210°

x = 70°.

The two smaller angles are both 70°.

The two larger angles are both 2(70°) - 30° = 110°.

Hence, option 2 is the correct option.

ABCD is a parallelogram in which AB = 9.5 cm and its perimeter is 30 cm. Length of each side of parallelogram ABCD is :

1. AB = 9.5 cm, DC = 9.5 cm, BC = 5.5 cm, DA = 5.5 cm

2. AB = 9.5 cm, BC = 9.5 cm, DC = 5.5 cm, DA = 5.5 cm

3. DC = 5.5 cm, BC = 5.5 cm, AB = 9.5 cm, AD = 9.5 cm

4. none of these

Answer

In a parallelogram, opposite sides are equal in length.

AB = CD

BC = DA

Perimeter = 30 cm

We know that,

Perimeter of Parallelogram = 2(Length + Breadth)

30 = 2(9.5 + BC)

15 = (9.5 + BC)

BC = 15 - 9.5

BC = 5.5 cm

BC = AD = 5.5 cm.

AB = CD = 9.5 cm

Hence, option 1 is the correct option.

Each side of a rhombus is 10 cm and one of its diagonals is 16 cm, length of other diagonal will be :

1. 11 cm

2. 12 cm

3. 13 cm

4. 15 cm

Answer

Let ABCD be the rhombus and the diagonals intersect at point O.

Let diagonal AC = 16 cm.

We know that,

Diagonals of rhombus bisect each other at right angles.

∴ AO = OC = $\dfrac{AC}{2} = \dfrac{16}{2}$ = 8 cm and BO = OD = x cm (let).

In right angle triangle AOB,

By pythagoras theorem,

⇒ (Hypotenuse)² = (Perpendicular)² + Base²

⇒ AB² = AO² + OB²

⇒ 10² = 8² + x²

⇒ 100 = 64 + x²

⇒ x² = 100 - 64

⇒ x² = 36

⇒ x = $\sqrt{36}$ = 6 cm.

From figure,

⇒ BD = BO + OD = 6 + 6 = 12 cm.

Hence, option 2 is the correct option.

ABCD is a rhombus. If ∠A = 70°, then ∠CDB will be :

1. 45°

2. 65°

3. 55°

4. 75°

Answer

The sum of consecutive angles in a rhombus is 180°.

∠A + ∠D = 180°

70° + ∠D = 180°

∠D = 180° - 70°

∠D = 110°.

The diagonal BD bisects ∠D.

∠CDB = $\dfrac{∠D}{2}$

∠CDB = $\dfrac{110°}{2}$

∠CDB = 55°.

Hence, option 3 is the correct option.

In a parallelogram, an angle is $\dfrac{4}{5}$ th of its adjacent angle, then angles of the parallelogram are :

1. 80°, 100°, 80°, 100°

2. 70°, 110°, 70°, 110°

3. 60°, 120°, 60°, 120°

4. none of these

Answer

In a parallelogram, adjacent angles are supplementary.

Let one of the angle of parallelogram be x.

The adjacent angle is given as $\dfrac{4x}{5}$.

$\Rightarrow x + \dfrac{4x}{5} = 180° \\[1em] \Rightarrow \dfrac{5x + 4x}{5} = 180° \\[1em] \Rightarrow \dfrac{9x}{5} = 180° \\[1em] \Rightarrow 9x = 180° \times 5 \\[1em] \Rightarrow 9x = 900° \\[1em] \Rightarrow x = 100°.$

The adjacent angle = $\dfrac{4 \times 100°}{5}$ = 80°.

The angles opposite to 100° is also 100°.

The angle opposite to 80° is also 80°.

Hence, option 1 is the correct option.

The lengths of diagonals of a rhombus are 24 cm and 18 cm respectively, length of each side of the rhombus is :

1. 25 cm

2. 15 cm

3. 35 cm

4. 45 cm

Answer

The diagonals of a rhombus are 18 cm and 24 cm.

AC = 18 cm

Then, OA = OC = $\dfrac{18}{2}$ = 9 cm.

And, BD = 24 cm

Then, OB = OD = $\dfrac{24}{2}$ = 12 cm.

Since the diagonals of a rhombus bisect at 90°.

Applying pythagoras theorem in triangle AOB, we get :

⇒ AB² = OA² + OB²

⇒ AB² = (9)² + (12)²

⇒ AB² = 81 + 144

⇒ AB² = 225

⇒ AB = $\sqrt{225}$

⇒ AB = 15 cm.

Hence, option 2 is the correct option.

ABCD is a parallelogram in which ∠BAD = 60° and ∠BAC = 30°, then ∠CBD =

1. 45°

2. 60°

3. 70°

4. 80°

Answer

Given,

∠BAD = 60° and ∠BAC = 30°.

From figure,

∠CAD = ∠BAD - ∠BAC = 60° - 30° = 30°.

If diagonal of a parallelogram bisects a vertex angle, then it is a rhombus.

Thus, ABCD is a rhombus.

In rhombus adjacent angles are supplementary.

Thus,

∠BAD + ∠CBA = 180°

60° + ∠CBA = 180°

∠CBA = 180° - 60° = 120°.

Since, diagonals of rhombus bisect the vertex angle,

∴ ∠CBD = $\dfrac{∠CBA}{2} = \dfrac{120°}{2}$ = 60°.

Hence, option 2 is the correct option.

The diagonals of the rectangle ABCD intersect at O. If ∠OBC = 64°, then ∠OAB =

1. 64°

2. 32°

3. 26°

4. 36°

Answer

In a rectangle all angles are equal to 90°.

∠ABC = ∠OBC + ∠ABO

90° = 64° + ∠ABO

∠ABO = 90° - 64°

∠ABO = 26°.

The diagonals of a rectangle are equal in length and bisect each other. This means :

⇒ AC = BD

⇒ AO = BO

In an isosceles triangle AOB, the angles opposite the equal sides are equal.

∠OAB = ∠ABO = 26°.

Hence, option 3 is the correct option.

If ∠ADB of the rhombus ABCD is 30°, then ∠ACB =

1. 30°

2. 60°

3. 70°

4. 90°

Answer

In rhombus opposite sides are parallel. The diagonal BD acts as a transversal line.

Therefore, the alternate interior angles are equal.

∠DBC = ∠ADB = 30°

From figure,

∠OBC = ∠DBC = 30°

∠BOC = 90° [Diagonals of rhombus cut at right angles]

In triangle BOC,

∠OBC + ∠BOC + ∠OCB = 180°

30° + 90° + ∠OCB = 180°

120° + ∠OCB = 180°

∠OCB = 180° - 120°

∠OCB = 60°.

Hence, option 2 is the correct option.

In the parallelogram ABCD, ∠A : ∠B = 3 : 5. ∠C =

1. 67.5°

2. 112.5°

3. 45°

4. 135°

Answer

Given the ratio ∠A : ∠B = 3 : 5.

Let ∠A = 3x and ∠B = 5x.

Sum of adjacent angles in a parallelogram = 180°.

3x + 5x = 180°

8x = 180°

x = 22.5°

In a parallelogram, opposite angles are equal.

∠C = ∠A = 3x

∠C = 3 × (22.5°)

∠C = 67.5°

Hence, option 1 is the correct option.

The bisectors of ∠A and ∠B of the parallelogram ABCD intersect at E. ∠AEB =

1. 100°

2. 90°

3. 80°

4. 60°

Answer

In a parallelogram, consecutive angles are supplementary.

∠A + ∠B = 180°

In triangle AEB,

∠AEB + ∠BAE + ∠EBA = 180°

∠AEB + $\dfrac{1}{2}$ ∠A + $\dfrac{1}{2}$ ∠B = 180°

∠AEB + $\dfrac{1}{2}$ (∠A + ∠B) = 180°

∠AEB + $\dfrac{1}{2} \times$ (180°) = 180°

∠AEB + 90° = 180°

∠AEB = 180° - 90°

∠AEB = 90°.

Hence, option 2 is the correct option.

The bisectors of ∠A and ∠B of the parallelogram ABCD intersect at P on the side CD. If BC = 3 cm, then AB =

1. 4 cm

2. 5 cm

3. 6 cm

4. 8 cm

Answer

In parallelogram ABCD, we know that AB ∥ DC. The line AP is a transversal.

∠PAB = ∠APD (Alternate interior angles are equal) ....(1)

Since AP is the bisector of ∠A:

∠PAB = ∠PAD .......(2)

From equation (1) and (2), we get :

∴ ∠PAD = ∠APD

In triangle APD, since two angles are equal, the triangle is isosceles.

AD = DP

ABCD is a parallelogram, AD = BC = 3 cm.

Thus, DP = 3 cm.

Similarly, for the bisector BP and transversal BP:

∠PBA = ∠BPC (Alternate interior angles are equal) .......(3)

Since BP bisects ∠B:

∠PBA = ∠PBC ..........(4)

From equation (3) and (4), we get :

∴ ∠BPC = ∠PBC

In triangle BCP, since two angles are equal, the triangle is isosceles.

PC = BC = 3 cm.

Length of CD = DP + PC = 3 + 3 = 6 cm.

AB = CD = 6 cm [Opposite sides of parallelogram are equal]

Hence, option 3 is the correct option.

If the lengths of the diagonals of a rhombus are 24 cm and 10 cm, then the length of its each side is :

1. 26 cm

2. 17 cm

3. 15 cm

4. 13 cm

Answer

Let the rhombus be ABCD and its diagonals AC and BD intersect at point O.

We know that,

Diagonals of a rhombus bisect each other.

AC = 24 cm

OA = $\dfrac{24}{2}$ = 12 cm.

BD = 10 cm

OB = $\dfrac{10}{2}$ = 5 cm.

In triangle AOB,

OA² + OB² = AB²

12² + 5² = AB²

144 + 25 = AB²

169 = AB²

AB = $\sqrt{169}$

AB = 13 cm.

Hence, option 4 is the correct option.

If the opposite angles of a quadrilateral are equal, then it will definitely be a :

1. rectangle

2. square

3. rhombus

4. parallelogram

Answer

If the opposite angles of a quadrilateral are equal, then it will definitely be a parallelogram.

Hence, option 4 is the correct option.

The diagonals of a quadrilateral are equal and they bisect each other. The quadrilateral is definitely a :

1. rectangle

2. square

3. rhombus

4. parallelogram

Answer

A rectangle is defined as a parallelogram with equal diagonals. If the diagonals of a quadrilateral bisect each other and are also equal, it must be a rectangle.

Hence, option 1 is the correct option.

PQRS is a rhombus in which PQ = 6 cm and ∠PQR = 120°. The length of the diagonal QS is :

1. 6 cm

2. 7 cm

3. 8 cm

4. 11 cm

Answer

In triangle PQS,

PQ = PS = 6 cm (because all sides of a rhombus are equal).

Therefore, the base angles are equal:

∠PQS = ∠PSQ.

∠PQR = ∠PSR = 120° [Opposite angles of a rhombus are equal]

∠PQR + ∠QPS = 180° [Consecutive angles are supplementary]

∠QPS = 180° - 120°

∠QPS = 60°.

In triangle PQS,

∠QPS + ∠PQS + ∠PSQ = 180°

60° + 2∠PQS = 180°

2∠PQS = 180° - 60°

2∠PQS = 120°

∠PQS = 60°.

Since all three angles (∠QPS, ∠PQS, and ∠PSQ) are 60°, △PQS is an equilateral triangle.

PQ = PS = QS = 6 cm.

Hence, option 1 is the correct option.