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. 24.6^2+ 29.7^2 ≠ 30.0^2 Let's find the measures of ∠ J, ∠ K, and ∠ L one at a time.

Finding m ∠ J

The lengths of all three sides of the triangle are given. Therefore, we can use the Law of Cosines to find m ∠ J. j^2=k^2+l^2 -2 k l cos J Let's substitute j= 29.7, k= 30.0, and l= 24.6 to isolate cos J.

Now, we can use the inverse cosine ratio and a calculator to find m ∠ J.

Finding m ∠ K

Now that we know the measure of ∠ J, we can find m ∠ K using the Law of Sines. sin J/j =sin K/k Let's substitute j= 29.7, m ∠ J ≈ 65, and k= 30, to isolate sin K.

Now we can use the inverse sine ratio to find m ∠ K.

Finding m ∠ L

Finally, to find m ∠ 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. 65+ 66 + m ∠ L = 180 ⇔ m ∠ L ≈ 49

Completing the Triangle

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