Exercise 23 Page 963

First, let's draw the measurements from the exercise on a right triangle to visualize the given information.

We will find the missing measures one at a time. In this case, this means that we want to find m∠ D, m∠ E, and d.

Angle Measures

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=21/51 By the definition of the inverse cosine, the inverse cosine of 2151 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. 65.7^(∘) +m∠ E≈ 90^(∘) ⇔ m∠ E≈ 24.3^(∘)

Side Lengths

Finally, we can find the measure of d. To do it we can use the Pythagorean Theorem. d^2+e^2=f^2 Let's substitute the known lengths, e= 21 and f= 51, into this equation to find d.