Using the Linear Pair Conjecture and the Triangle Exterior Angle Theorem, write two equation for the acute angles.
40^(∘) and 50^(∘)
Practice makes perfect
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.
The measure of each exterior angle of a triangle
equals the sum of the measures of
its two remote interior angles.
We can illustrate this theorem as shown below.
m∠ 1= m∠ 2+ m∠ 3
Having used the theorem, we can write the exterior angles as the following.
Acute Angle I: & (90+x)^(∘)
Acute Angle II: & (90+y)^(∘)
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.
Equation I: 90+x+y=180
We will write the second equation by using the given ratio.
Equation II: 90+x/90+y=13/14
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.
