Exercise 33 Page 594

Let's consider the given diagram. We will use the same color for a side and its opposite vertex. This will help us use the Law of Cosines and later the Law of Sines.

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

Finding $m\angle J$

The lengths of all three sides of the triangle are given. Therefore, we can use the Law of Cosines to find $m\angle J.$ Let's substitute $j=29.7,$ $k=30.0,$ and $\ell =24.6$ to isolate $\cos J.$ Now, we can use the inverse cosine ratio and a calculator to find $m\angle J.$

Finding $m\angle K$

Now that we know the measure of $\angle J,$ we can find $m\angle K$ using the Law of Sines. Let's substitute $j=29.7,$ $m\angle J\approx 65,$ and $k=30,$ to isolate $\sin K.$ Now we can use the inverse sine ratio to find $m\angle K.$

Finding $m\angle L$

Finally, to find $m\angle L$ 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.