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
1^{st} angle = 5(15) = 75°
2^{nd} angle = 4(15) = 60°
3^{rd} 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
1^{st} angle = 3(12) = 36°
2^{nd} angle = 4(12) = 48°
3^{rd} 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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