HOW TO FIND ANGLES OF A TRIANGLE WITH RATIO

The following steps would be useful to find the angles of a triangle from the given ratio. 

Step 1 : 

Let the angles of a triangle are in the ratio a : b : c. To get three angles, multiply each term of the ratio by an unknown, say 'x'. 

Then, the three angles are ax, bx and cx.

Step 2 : 

Since the angles of a triangle add up to 180°, 

ax + bx + cx = 180

Solve for x in the above equation and multiply the value of x by a, b and c separately to find the measure of each angle. 

Example 1 :

If the angles of a triangle are in the ratio 5 : 4 : 3, then find the measure of each angle.  

Solution :

From the ratio 5 : 4 : 3, the angles of the triangle are 

5x, 4x and 3x

Sum of the angles of a triangle = 180°

5x + 4x + 3x = 180

Simplify.

12x = 180

Divide each side by 12. 

x = 15

1st angle = 5(15) = 75°

2nd angle = 4(15) = 60°

3rd angle = 3(15) = 45°

Example 2 :

If the angles of a triangle are in the ratio 3 : 4 : 8, then find the measure of each angle.  

Solution :

From the ratio 3 : 4 : 8, the angles of the triangle are 

3x, 4x and 8x

Sum of the angles of a triangle = 180°

3x + 4x + 8x = 180

Simplify. 

15x = 180

Divide each side by 15.

x = 12

1st angle = 3(12) = 36°

2nd angle = 4(12) = 48°

3rd angle = 8(12) = 96°

Example 3 :

In a right triangle ABC, angle A is right angle and the ratio between the angles B and C is 2 : 3. Find the measures of angle B and C.  

Solution :

From the ratio 2 : 3, the angle B and C are 2x and 3x. 

Sum of the angles of a triangle = 180°

m∠A + m∠B + m∠C = 180°

Substitute. 

90 + 2x + 3x = 180

Simplify.

90 + 5x = 180

Subtract 90 from each side.

5x = 90

Divide each side by 5.

x = 18

m∠B = 2(18) = 36°

m∠C = 3(18) = 54°

Example 4 :

In a triangle ABC, measure of A is one of the measure of B and the ratio between the measures of B and C is 2 : 3. Find the measure of each angle.   

Solution :

Given : Measure of angle A is one of the measure of angle B.

A = (1/2)B

∠A/∠B = 1/2

∠A : ∠B = 1 : 2 ----(1)

Given : Measures of B and C is 2 : 3. Find the measure of each angle.   

∠B : ∠C = 2 : 3 ----(3)

From (1) and (2), ∠A, ∠B and ∠C are in the ratio 1 : 2 : 3.

From the ratio 1 : 2 : 3, the measures ∠A, ∠B and ∠C are 

x, 2x and 3x

Sum of the angles of a triangle = 180°

x + 2x + 3x = 180

Simplify. 

6x = 180

Divide each side by 6.

x = 30

∠A = 30°

∠B = 2(30) = 60°

∠C = 3(30) = 90°

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