Exercise 35 Page 177

We have been told that a right triangle has exterior angles at each of its acute angles with measures in the ratio $13:14.$ We will first draw the right triangle. Let $x$ and $y$ be the measures of the acute angles.

Next, we will determine the exterior angles at each of its acute angles. To do that, we will use the Triangle Exterior Angle Theorem. We can illustrate this theorem as shown below. Having used the theorem, we can write the exterior angles as the following. Let's place the exterior angles on the triangle for a better understanding. We know that an interior angle and its exterior angle form a linear pair. Thus, our first equation is the following. We will write the second equation by using the given ratio. Finally, we will solve these equations together and find the measures of the acute angles. Let's start with using cross product to rewrite Equation II. Thus, the measure $x$ is $40^{\circ }.$ Next, we will find the measure of $y.$ As a result, the measure $y$ is $50^{\circ }.$