# Questions on triangles & its angles

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.

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