Answer
$D$
Work Step by Step
We are given the lengths of three sides of a triangle, and we need to find the measure of one angle. We can use the Law of Cosines in this case.
$12^2 = 10^2 + 8^2 - 2(10)(8)$ cos $\angle Z$
Evaluate exponents first, according to order of operations:
$144 = 100 + 64 - 2(10)(8)$ cos $\angle Z$
Add to simplify on the right side of the equation:
$144 = 164 - 2(10)(8)$ cos $\angle Z$
Multiply to simplify:
$144 = 164 - 160$ cos $\angle Z$
Subtract $164$ from each side of the equation to move constants to the left side of the equation:
$-20 = -160$ cos $\angle Z$
Divide each side by $-160$:
cos $\angle Z = \frac{-20}{-160}$
Take $cos^{-1}$ to solve for $\angle Z$:
$m \angle Z \approx 82.8^{\circ}$
The answer is option $D$.
