Answer
$m \angle x \approx 46.8^{\circ}$
$m \angle y \approx 35.0^{\circ}$
Work Step by Step
First, we want to know what side is opposite the angle in question. The side that is opposite to the angle we are looking for, $\angle x$, has a measure of $14$, so let's plug in what we know into the formula for the law of cosines:
$14^2 = 11^2 + 19^2 - 2(11)(19)$ cos $\angle x$
Evaluate exponents first, according to order of operations:
$196 = 121 + 361 - 2(11)(19)$ cos $\angle x$
Add to simplify on the right side of the equation:
$196 = 482 - 2(11)(19)$ cos $\angle x$
Multiply to simplify:
$196 = 482 - 418$ cos $\angle x$
Subtract $418$ from each side of the equation to move constants to the left side of the equation:
$-286 = -418$ cos $m \angle x$
Divide each side by $-418$:
cos $m \angle x = \frac{-286}{-418}$
Take $cos^{-1}$ to solve for $\angle x$:
$m \angle x \approx 46.8^{\circ}$
Now, we use the law of cosines to find $m \angle y$. The line opposite to $\angle y$ measures $11$, so now, we can set up the equation:
$11^2 = 14^2 + 19^2 - 2(14)(19)$ cos $\angle y$
Evaluate exponents first, according to order of operations:
$121 = 196 + 361 - 2(14)(19)$ cos $\angle y$
Add to simplify on the right side of the equation:
$121 = 557 - 2(14)(19)$ cos $\angle y$
Multiply to simplify:
$121 = 557 - 532$ cos $\angle y$
Subtract $557$ from each side of the equation to move constants to the left side of the equation:
$-436 = -532$ cos $m \angle y$
Divide each side by $-532$:
cos $m \angle y = \frac{-436}{-532}$
Take $cos^{-1}$ to solve for $\angle y$:
$m \angle y \approx 35.0^{\circ}$
