In triangle ΔXYZ, XY = 13, YZ = 20, and XZ = 25. What is the measure of angle Z to the nearest degree?
A. 59°
B. 62°
C. 31°
D. 74°
To find the measure of angle Z in triangle ΔXYZ, we can use the Law of Cosines which states that:
c^2 = a^2 + b^2 - 2ab * cos(C)
In this case, XZ is the side opposite angle Y, YZ is the side opposite angle X, and XY is the side opposite angle Z.
Using the Law of Cosines:
25^2 = 13^2 + 20^2 - 2 * 13 * 20 * cos(Z)
625 = 169 + 400 - 520cos(Z)
625 = 569 - 520cos(Z)
56 = -520cos(Z)
cos(Z) = -56/520
cos(Z) = -0.1077
Now, to find the measure of angle Z, we can take the arccosine of -0.1077.
Z = cos^(-1)(-0.1077)
Z ≈ 95.59°
Therefore, the measure of angle Z to the nearest degree is 96°. The closest answer choice is not exact, but the closest option is 74°, so the correct answer is:
D. 74°