Triangles — Multiple Choice Questions

The value of angle x is :

1. 30°

2. 50°

3. 40°

4. 80°

Answer

In the figure,

∠BAC = 80° (Vertically opposite angles are equal)

In triangle ABC, by the angle sum property,

⇒ 80° + 50° + x = 180°

⇒ 130° + x = 180°

⇒ x = 180° − 130°

⇒ x = 50°

Hence, option 2 is the correct option.

The value of angle x is :

1. 84°

2. 96°

3. 132°

4. 48°

Answer

From figure,

∠BAC = 84° (Vertically opposite angles are equal)

Since ABC is an isosceles triangle, its base angles are equal.

∴ ∠ABC = ∠ACB

By angle sum property,

⇒ ∠BAC + ∠ABC + ∠ACB = 180°

⇒ ∠BAC + 2∠ACB = 180°

⇒ 84° + 2∠ACB = 180°

⇒ 2∠ACB = 180° − 84°

⇒ 2∠ACB = 96°

⇒ ∠ACB = $\dfrac{96\degree}{2}$ = 48°

As x and ∠ACB form a linear pair,

⇒ x + ∠ACB = 180°

⇒ x + 48° = 180°

⇒ x = 180° − 48°

⇒ x = 132°

Hence, option 3 is the correct option.

One angle of a triangle is 60° and the other two angles are in the ratio 1 : 2; the bigger of these two angles is :

1. 60°

2. 90°

3. 40°

4. 80°

Answer

Let the other two angles be x and 2x.

⇒ 60° + x + 2x = 180°

⇒ 60° + 3x = 180°

⇒ 3x = 180° − 60°

⇒ 3x = 120°

⇒ x = $\dfrac{120\degree}{3}$

⇒ x = 40°

∴ The two angles are 40° and 80°, and the bigger one is 80°.

Hence, option 4 is the correct option.

The angles of a triangle are in the ratio 2 : 3 : 1; the smallest angle is :

1. 30°

2. 45°

3. 60°

4. none of these

Answer

Let the angles be 2x, 3x and x.

⇒ 2x + 3x + x = 180°

⇒ 6x = 180°

⇒ x = $\dfrac{180\degree}{6}$

⇒ x = 30°

∴ The smallest angle is x = 30°.

Hence, option 1 is the correct option.

In the given figure, DE is parallel to BC. The value of x is :

1. 50°

2. 45°

3. 40°

4. 35°

Answer

∠AED and ∠CED form a linear pair, so ∠AED = 180° − 95° = 85°.

Since DE ∥ BC, ∠ADE = ∠ABC = 50° (corresponding angles).

In ΔADE, by the angle sum property,

⇒ x + 85° + 50° = 180°

⇒ x + 135° = 180°

⇒ x = 180° − 135°

⇒ x = 45°

Hence, option 2 is the correct option.

The value of angle x is :

1. 76°

2. 68°

3. 56°

4. none of these

Answer

In the triangle ABC, the two marked sides CA and CB are equal, so the base angles ∠CAB and ∠CBA are equal.

Since ∠ABD and ∠CBA are on the same line, they form a linear pair.

⇒ ∠ABD + ∠CBA = 180°

⇒ 124° + ∠CBA = 180°

⇒ ∠CBA = 180° − 124°

⇒ ∠CBA = 56°

⇒ ∠CAB = ∠CBA = 56°

By the angle sum property,

⇒ ∠ACB + ∠CBA + ∠CAB = 180°

⇒ x + 56° + 56° = 180°

⇒ x + 112° = 180°

⇒ x = 180° − 112°

⇒ x = 68°

Hence, option 2 is the correct option.

The value of angle x is :

1. 45°

2. 25°

3. 30°

4. none of these

Answer

In △ABC, AB = AC

⇒ ∠B = ∠ACB

As ∠BAC = 90°,

By angle sum property,

⇒ ∠B + ∠ACB + ∠BAC = 180°

⇒ 2∠B + 90° = 180°

⇒ ∠B = $\dfrac{180\degree − 90\degree}{2}$ = 45°

∠ACB + ∠ACD = 180°

45° + ∠ACD = 180°

∠ACD = 135°

In △ACD, AC = CD

⇒ ∠D = ∠CAD = x

By angle sum property,

⇒ ∠D + ∠ACD + ∠CAD = 180°

⇒ x + 135° + x = 180°

⇒ 2x = 180° − 135° = 45°

⇒ x = $\dfrac{45\degree}{2}$ = 22·5° = 22°30′

Hence, option 4 is the correct option.

The length of AB is :

1. 23 unit

2. 46 unit

3. 61 unit

4. 73 unit

Answer

The two sides marked equal are AC and BC, so

⇒ x + 38 = 2x + 15

⇒ 38 − 15 = 2x − x

⇒ 23 = x

⇒ x = 23

From the figure, AB = 3x + 4.

Substituting x = 23,

⇒ AB = 3 × 23 + 4

⇒ AB = 69 + 4

⇒ AB = 73 unit

Hence, option 4 is the correct option.

The value of angle x is :

1. 35°

2. 45°

3. 55°

4. 65°

Answer

In △ADB, by angle sum property:

⇒ ∠A + ∠B + ∠D = 180°

⇒ 50° + ∠B + 90° = 180°

⇒ ∠B = 180° − 140° = 40°

In △BCE, by angle sum property:

⇒ ∠E + ∠B + ∠C = 180°

⇒ 85° + 40° + x = 180°

⇒ 125° + x = 180°

⇒ x = 180° − 125° = 55°

Hence, option 3 is the correct option.

The value of angle x is :

1. 54°

2. 126°

3. 108°

4. 306°

Answer

In isosceles triangle ADE,

DA = DE

⇒ ∠DEA = ∠DAE [Base angles of an isosceles triangle are equal]

By angle sum property,

⇒ ∠DAE + ∠DEA + ∠ADE = 180°

⇒ ∠DAE + ∠DAE + ∠ADE = 180°

⇒ 2∠DAE + 72° = 180°

⇒ ∠DAE = $\dfrac{180\degree − 72\degree}{2} = 54\degree$

Since DE is parallel to BC,

⇒ ∠B = ∠ADE = 72° (corresponding angles are equal)

Now, in triangle ABC,

By angle sum property,

⇒ ∠A + ∠B + ∠ACB = 180°

⇒ 54° + 72° + ∠ACB = 180°

⇒ ∠ACB = 180° − 126° = 54°

As, x is the reflex angle, so

⇒ x = 360° − ∠ACB = 360° − 54° = 306°

Hence, option 4 is the correct option.