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Find triangle angles in the ratio 1 to 3 to 5

The interior angles of a triangle add to 180 degrees

When triangle angles are given as multiples of one variable, add their expressions and set the total equal to $18 0^{\circ}$ . Solve for the variable, then find each actual angle.

Problem

In triangle $P QR$ , $m ∠ P = x^{\circ}$ , $m ∠ Q = (3 x)^{\circ}$ , and $m ∠ R = (5 x)^{\circ}$ . Find $x$ and the three angles.

Answer

x = 20

. The angles are

2 0^{\circ}

6 0^{\circ}

, and

10 0^{\circ}

The variable x gives the smallest angle; it is not the value of all three angles.

Step-by-step solution

1. Write the triangle angle sum

The sum of the three interior angles is $18 0^{\circ}$ , so $x + 3 x + 5 x = 180$ .

2. Solve for x

Combine terms to get $9 x = 180$ . Dividing by 9 gives $x = 20$ .

3. Find and check the angles

The angles are $x = 2 0^{\circ}$ , $3 x = 6 0^{\circ}$ , and $5 x = 10 0^{\circ}$ . Their sum is $18 0^{\circ}$ , and their ratio is $1 : 3 : 5$ .

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