Exercise 31 Page 827

Let's begin by color coding the opposite angles and sides in the given triangle. It will help us use the Law of Sines later.

Let's find the length of c and the measures of ∠ B, ∠ C one at a time.

Finding m ∠ B

The measures of all two sides and the angle opposite one of them are given. Therefore, we can use the Law of Sines to find m ∠ B. sin A/a = sin B/b Let's substitute A= 104^(∘), a= 12.4, and b= 8.1 to isolate sin B.

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

Finding m ∠ C

To find m ∠ C we can use the Triangle Angle Sum Theorem. This tells us that the measures of the angles in a triangle add up to 180^(∘). 104^(∘) + 39^(∘) + m∠ C = 180^(∘) ⇕ m ∠ C ≈ 37^(∘)

Finding c

Finally, we can find c by using the Law of Sines. sin A/a = sin C/c Let's substitute A= 104^(∘), a= 12.4, and C= 37^(∘) to isolate c.

Completing the Triangle

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

Extra

The Law of Sines can be visualized by drawing a triangle and labeling angles and sides.