In triangle ΔXYZ, XY = 13, YZ = 20, and XZ = 25. What is the measure of angle Z to the nearest degree?

Question

(1 point)
Responses

59°
59°

62°
62°

31°
31°

74°

Answer 1

To find the measure of angle Z, we can use the Law of Cosines.

The formula is as follows: c^2 = a^2 + b^2 - 2abcos(C), where c is the side opposite angle C.

In this case, side XY is opposite angle Z, so we can write:
25^2 = 13^2 + 20^2 - 2(13)(20)cos(Z)

Simplifying this equation gives us:
625 = 169 + 400 - 520cos(Z)

Combining like terms:
456 = -520cos(Z)

Dividing both sides by -520:
cos(Z) = -456/520

Taking the inverse cosine of both sides:
Z = arccos(-456/520)

Using a calculator to find the arccos of -456/520 gives us a value of approximately 31 degrees.

Therefore, the measure of angle Z to the nearest degree is 31°.