Given below are collection of questions related to triangle and its angles with detailed solution.

All the questions are to the standard of grade 09.

**Question 01**

Given below is triangle ABC with ∠A = 55 and ∠B = 40 degree. Find the measure of ∠C.

**Solution**

We know that sum of all internal angle of triangle measure 180 degree.

∠A + ∠B + ∠C = 180

55 + 40 + ∠C = 180

95 + ∠C = 180

∠C = 180 – 95

∠C = 85 degree

Hence, ∠C measures 85 degree.

**Question 02**

In a triangle, the angles are in the ratio 1 : 2 : 3. Find the measure of all the angles.

**Solution**

Let the measure of angles are x, 2x and 3x.

We know that sum of all internal angle of triangle measure 180 degree.

x + 2x + 3x = 180

6x = 180

x = 60

The values of all angles are;

x = 60 degree

2x = 2(60) = 120 degree

3x = 3 (60) = 180 degree

**Question 03**

The measure of angle of triangle is given as (x – 40), (x – 20) and (x/2 – 10). Find the exact value of all the angles.

**Solution**

We know that sum of internal angle of triangle measure 180 degree.

(x – 40) + (x – 20) + (x/2 – 10) = 180

2x + x/2 – 70 = 180

5x/2 = 180 + 70

5x/2 = 250

x = 100

We got the value of value of x, now let’s calculate the value of all the angles.

(x – 40) = 100 – 40 = 60 degree

(x – 20) = 100 – 20 = 80 degree

(x/2 -10) = 100/2 – 10 = 40 degree

**Question 04**

In a triangle two of the angles are equal and the third angle is greater than other angles by 30 degree. Find the measure of all angles.

**Solution**

Let the three angles are x, x and x + 30

Now using angle sum property of triangle, we can write;

x + x + x + 30 = 180

3x + 30 = 180

3x = 180 – 30

3x = 150

x = 50

We got the value of x, now let’s calculate the value of all angles.

x = 50 degree

x + 30 = 50 + 30 = 80 degree

The value of all the angles are 50, 50 and 80 degrees.

**Question 05**

In a triangle, the measure of one angle is equal to sum of other two angles. Prove that the given triangle is right angle triangle.

**Solution**

Since, one angle is equal to sum of other two angle, we can write following expression.

∠A = **∠**B + ∠C

We know that sum of all internal angle of triangle measures 180 degree.

∠A + ∠B + ∠C = 180

∠A + ∠A = 180

2∠A = 180

∠A = 90 degree

Since one angle of triangle measures 90 degrees, it means that the given triangle is a right triangle.

**Question 06**

Consider triangle ABC in which ∠A = 72 degree. Here line OB & OC are angle bisectors and they meet at point O. Find the measure of ∠BOC

**Solution**

Applying angle sum property in triangle ABC.

∠A + ∠B + ∠C = 180

72 + 2(∠OBC) + 2 (∠OCB) = 180

2(∠OBC) + 2 (∠OCB) = 180 – 72

2(∠OBC) + 2 (∠OCB) = 108

∠OBC + ∠OCB = 54 degree

Now apply angle sum property in triangle OBC.

∠OBC + ∠OCB + ∠BOC = 180

54 + ∠BOC = 180

∠BOC = 180 – 54

∠ BOC = 126 degree

Hence, value of ∠BOC is 126 degree.

**Question 07**

The below triangle ABC satisfy the relation B – A = C – B. Find the measurement of angle B.

**Solution**

Solving the given relation;

B – A = C – B

B + B = C + A

2B = C + A

Now apply angle sum property of triangle.

A + B + C = 180

B + 2B = 180 { using above expression 2B = C + A }

3B = 180

B = 60 degree

Hence, angle B measures 60 degree.

**Question 08**

Consider the triangle ABC in which ∠B + ∠C = 100 degree. Also line OB & OC bisect ∠B & ∠C respectively. Find the measure of ∠BOC

**Solution**

Since OB & OC are angle bisector of ∠B & ∠C, we can write;

∠OBC = 1/2 (∠B)

∠ OCB = 1/2 (∠C)

Applying angle sum property in triangle OBC.

∠OBC + ∠OCB + ∠BOC = 180

1/2 (∠B) + 1/2 (∠C) + ∠BOC = 180

1/2 ( ∠B + ∠C) + ∠BOC = 180

1/2 (100) + ∠BOC = 180

50 + ∠BOC = 180

∠BOC= 180 – 50

∠BOC = 130

Hence, ∠BOC measures 130 degree.