# A right triangle has exterior angles at each of its acute angles with measures in the ratio 13 14. How do you find the measures of the two acute angles of the right triangle?

Jul 13, 2017

56.67° and 33.33°

#### Explanation:

Say that $x$ is the measure of one of the acute angles. The measure of the exterior angle would be $360 - x$.

Since this is a right triangle, the measure of the two acute angles must add up to 90°. Thus, we can say the other acute angle is equal to $90 - x$. The exterior angle is $360 - \left(90 - x\right)$, which can be simplified to $270 + x$.

Thus, the measures of the two exterior angles are $360 - x$ and $270 + x$. We can now write a proportion:

$\frac{360 - x}{270 + x} = \frac{13}{14}$

$14 \left(360 - x\right) = 13 \left(270 + x\right)$ $\to$ cross-multiply

$5040 - 14 x = 3510 + 13 x$ $\to$ distribute

$1530 = 27 x$ $\to$ add $14 x$ and subtract $3510$ from both sides

x~~56.67° $\to$ divide both sides by $27$

The other angle is 90° - 56.67° ~~ 33.33°.

The two acute angles are 56.67° and 33.33°.