First, let's draw the measurements from the exercise on a right triangle to visualize the given information. Note that it might not be to scale.
We will find the missing measures one at a time. In this case, this means that we want to find f, m∠D, and m∠E.
Hypotenuse
We can find the hypotenuse f using the Pythagorean Theorem.
d^2+ e^2= f^2
Let's substitute the known lengths, d = 10 and e= 12, into this equation to find f.
We can find m∠D using the cosine ratio. The cosine of ∠D is the ratio of the length of the leg adjacent ∠D to the length of the hypotenuse.
cos D=Adjacent/Hypotenuse ⇒ cos D=12/15.6
By the definition of the inverse cosine, the inverse cosine of 1215.6 is the measure of ∠D. To find it we can use a calculator.
To find m∠E, recall that the acute angles of a right triangle are complementary. Therefore, m∠D and m∠E add to 90^(∘).
m∠D+m∠E=90^(∘)
Now, we can substitute the approximated measure of ∠D in our equation and find the measure of ∠E.
39.7^(∘) +m∠E≈ 90^(∘) ⇔ m∠E≈ 50.3^(∘)
