The measures of a triangle's three interior angles are in the ratio $1: 4: 7$. What are the measures of the angles?
(A) $12^{\circ}, 48^{\circ}, 120^{\circ}$
(D) $15^{\circ}, 60^{\circ}, 105^{\circ}$
(B) $10^{\circ}, 40^{\circ}, 70^{\circ}$
(C) $20^{\circ}, 25^{\circ}, 155^{\circ}$
Solution 1

The sum of the three interior angles in a triangle is $180^{\circ}$.

A triangle's three interior angles are in the ratio $1: 4: 7$ and so the smallest of the angles measures $\frac{1}{1+4+7}=\frac{1}{12}$ of the sum of the three interior angles.

Thus, the smallest angle in the triangle meausres $\frac{1}{12}$ of $180^{\circ}$ or $\frac{180^{\circ}}{12}=15^{\circ}$.

The measure of the next largest angle is 4 times the measure of the smallest angle or $4 \times 15^{\circ}=60^{\circ}$.

The measure of the largest angle is 7 times the measure of the smallest angle or $7 \times 15^{\circ}=105^{\circ}$. The measures of the interior angles are $15^{\circ}, 60^{\circ}$ and $105^{\circ}$.

## Solution 2

The sum of the three interior angles in a triangle is $180^{\circ}$.

Working backward from the possible answers, we may eliminate (B) and (C) since the sum of the three given angles is not $180^{\circ}$.

The measures of the smallest and largest angles in the triangle are in the ratio $1: 7$.

Since $7 \times 12^{\circ}=84^{\circ}$ and not $120^{\circ}$, we may eliminate (A).

Since $7 \times 14^{\circ}=98^{\circ}$ and not $110^{\circ}$, we may eliminate (E).

The remaining possibility is (D) and we may confirm that $15^{\circ}, 60^{\circ}$ and $105^{\circ}$ have a sum of $180^{\circ}$ and are in the ratio $1: 4: 7$.