Lines & Angles — Multiple Choice Questions

Two complementary angles are in the ratio 1 : 2, the smaller angle is :

1. 60°

2. 30°

3. 120°

4. 90°

Answer

As the angles are in the ratio 1 : 2, let the angles be x and 2x.

As the angles are complementary, their sum is 90°.

⇒ x + 2x = 90°

⇒ 3x = 90°

⇒ x = $\dfrac{90\degree}{3}$

⇒ x = 30°

So, the angles are 30° and 60°, and the smaller angle is 30°.

Hence, option 2 is the correct option.

Two angles 3x − 20° and 2x + 30° are supplementary, the value of x is :

1. 34

2. 38

3. 50

4. 10

Answer

As the angles are supplementary, their sum is 180°.

⇒ 3x − 20° + 2x + 30° = 180°

⇒ 5x + 10° = 180°

⇒ 5x = 170°

⇒ x = $\dfrac{170\degree}{5}$

⇒ x = 34°

Hence, option 1 is the correct option.

In the given figure, AOB is a straight line and y = 30°, the value of x is :

1. 20°

2. 70°

3. 30°

4. 60°

Answer

Given,

y = 30°

From the figure, AOB is a straight line.

⇒ 2y + 30° + 3x = 180°     [Angles on a straight line]

⇒ 2 × 30° + 30° + 3x = 180°

⇒ 90° + 3x = 180°

⇒ 3x = 90°

⇒ x = $\dfrac{90\degree}{3}$

⇒ x = 30°

Hence, option 3 is the correct option.

In the given figure, BA is parallel to DE, the ∠BCD is equal to :

1. 70°

2. 90°

3. 110°

4. none of these

Answer

Through the point C, draw a line CF parallel to BA and DE.

Since BA ∥ CF and BC is a transversal,

⇒ ∠BCF + ∠ABC = 180°     [Co-interior angles are supplementary]

⇒ ∠BCF + 100° = 180°

⇒ ∠BCF = 80°

Since, CF ∥ DE and CD is a transversal,

⇒ ∠FCD + ∠CDE = 180°     [Co-interior angles are supplementary]

⇒ ∠FCD + 150° = 180°

⇒ ∠FCD = 30°

⇒ ∠BCD = ∠BCF + ∠FCD = 80° + 30° = 110°

Hence, option 3 is the correct option.

In the given figure, AB is parallel to CD and PQ is a transversal. If ∠a = 3x - 30° and ∠b = 2x + 10°; then the value of x is :

1. 20°

2. 40°

3. 30°

4. 50°

Answer

Since AB ∥ CD and PQ is a transversal,

⇒ ∠a = ∠b     [Corresponding angles are equal]

⇒ 3x − 30° = 2x + 10°

⇒ 3x − 2x = 10° + 30°

⇒ x = 40°

Hence, option 2 is the correct option.

In the given figure, BA is parallel to CE. The angle ABC is :

1. 50°

2. 60°

3. 70°

4. 55°

Answer

Since BA ∥ CE and BD is a transversal,

⇒ ∠ABC = ∠ECD     [Corresponding angles are equal]

⇒ ∠ABC = 60°

Hence, option 2 is the correct option.

In the given figure, AB is parallel to CD. The value of x is :

1. 50°

2. 60°

3. 55°

4. 45°

Answer

Through the point O, draw a line EO parallel to AB and so parallel to CD.

Let the ray from O meet AB at P and CD at Q.

From the figure, the angle marked 130° and ∠OQD form a linear pair.

⇒ ∠OQD + 130° = 180°

⇒ ∠OQD = 180° − 130° = 50°

Since the line through O is parallel to CD and OQ is a transversal,

⇒ ∠EOQ = ∠OQD = 50°     [Interior alternate angles are equal]

Since the line OE is parallel to AB and OP is a transversal,

⇒ ∠EOP = x     [corresponding angles are equal]

Now, the angle at O = 95°

⇒ ∠EOP + ∠EOQ = 95°

⇒ x + 50° = 95°

⇒ x = 45°

Hence, option 4 is the correct option.

In the given figure, BA is parallel to DC and MN is a transversal, then ∠a is :

1. 55°

2. 125°

3. 120°

4. 60°

Answer

Since BA ∥ DC and MN is a transversal,

⇒ ∠CEO = ∠FED = 5x     [Vertically opposite angles are equal]

As, BA ∥ DC and MN is a transversal,

⇒ ∠CEO = ∠a     (Corresponding angles are equal)

⇒ ∠a = 5x

Also, ∠a and ∠AOE form a linear pair.

⇒ ∠a + 3x − 20° = 180°

⇒ 5x + 3x − 20° = 180°

⇒ 8x − 20° = 180°

⇒ 8x = 200°

⇒ x = $\dfrac{200\degree}{8}$

⇒ x = 25°

⇒ ∠a = 5x = 5 × 25° = 125°.

Hence, option 2 is the correct option.

In the given figure, AB ∥ CD and PQ ∥ RO, then ∠a is :

1. 68°

2. 112°

3. 158°

4. none of these

Answer

Since PQ ∥ RO and AB is a transversal,

⇒ ∠AOR = 68°     [Alternate interior angles are equal]

Since AB ∥ CD and RO is a transversal,

⇒ ∠a = ∠AOR     [Corresponding angles are equal]

⇒ ∠a = 68°

Hence, option 1 is the correct option.

The supplement of an angle is four times its complement. The angle is :

1. 30°

2. 40°

3. 60°

4. 36°

Answer

Let the angle be x.

Supplement of x = 180° − x and complement of x = 90° − x

As the supplement of the angle is four times its complement,

⇒ 180° − x = 4(90° − x)

⇒ 180° − x = 360° − 4x

⇒ 4x − x = 360° − 180°

⇒ 3x = 180°

⇒ x = $\dfrac{180\degree}{3}$

⇒ x = 60°

Hence, option 3 is the correct option.