Answer
$x \approx 54.1$
$m \angle y \approx 72.0^{\circ}$
Work Step by Step
First, we want to know what angle is opposite the side in question. The angle that is opposite to the side we are looking for, $x$, is the angle that measures $40^{\circ}$, so let's plug in what we know into the formula for the law of cosines:
$x^2 = 78^2 + 80^2 - 2(78)(80)$ cos $40^{\circ}$
Evaluate exponents first, according to order of operations:
$x^2 = 6084 + 6400 - 2(78)(80)$ cos $40^{\circ}$
Add to simplify on the right side of the equation:
$x^2 = 12484 - 2(78)(80)$ cos $40^{\circ}$
Take the square root of both sides of the equation:
$x \approx 54.1$
Now, we use the law of cosines to find $m \angle y$. The line opposite to $\angle y$ measures $80$, so now, we can set up the equation:
$80^2 = 54.1^2 + 78^2 - 2(54.1)(78)$ cos $\angle y$
Evaluate exponents first, according to order of operations:
$6400 = 2926.81 + 6084 - 2(54.1)(78)$ cos $\angle y$
Add to simplify on the right side of the equation:
$6400 = 9010.81 - 2(54.1)(78)$ cos $\angle y$
Multiply to simplify:
$6400 = 9010.81 - 8439.6$ cos $\angle y$
Subtract $9010.81$ from each side of the equation to move constants to the left side of the equation:
$-2610.81 = -8439.6$ cos $\angle y$
Divide each side by $-30$:
cos $\angle y = \frac{-2610.81}{-8439.6}$
Take $cos^{-1}$ to solve for $\angle y$:
$m \angle y \approx 72.0^{\circ}$
