Exercise 11 Page 592

Let's begin by drawing $△DEF$ and labeling the lengths of the sides. We will also color code the opposite angles and sides. It will help us use the Law of Sines and Law of Cosines later.

First, we can tell that it is not a right triangle, as the sides do not satisfy the Pythagorean Theorem. Let's find the measures of $\angle D,$ $\angle E,$ and $\angle F$ one at a time.

Finding $m\angle D$

The measures of all three sides of the triangle are given. Therefore, we can use the Law of Cosines to find $m\angle D.$ Let's substitute $d=21.6,$ $e=20,$ and $f=16$ to isolate $\cos D.$ Now, we can use the inverse cosine ratio and a calculator to find $m\angle D.$

Finding $m\angle E$

Now that we know the measure of $\angle D,$ we can find $m\angle E$ using the Law of Sines. Let's substitute $d=21.6,$ $e=20,$ and $m\angle D\approx 73$ to isolate $\sin E.$ Now we can use the inverse sine ratio to find $m\angle E.$

Finding $m\angle F$

Finally, to find $m\angle F$ we can use the Triangle Angle Sum Theorem. This tells us that the measures of the angles in a triangle add up to $180.$

Completing the Triangle

With all of the angle measures, we can complete our diagram.