Answer
$w = 3$
$x = 41^{\circ}$
Work Step by Step
Since we have the measure of one angle and the measure of the hypotenuse, we can use the sine ratio to find $w$, which is the side opposite to the given angle.
The sine ratio is given as follows:
sin $A = \frac{opposite}{hypotenuse}$
Let's plug in what we know:
$\frac{x}{y} = \frac{w}{6}$
Multiply each side by $6$:
$w$ = sin $30^{\circ}(6)$
Find sin $30^{\circ}$:
$w = (0.5)(6)$
Multiply to solve:
$w = 3$
We have another right triangle because we have a transversal cutting two parallel lines, meaning alternate interior angles are congruent. One of the alternate interior angles is a right angle in a right triangle; therefore, the other alternate interior angle is also a right angle in a triangle.
In this other right triangle, we just found the measure of $w$, which is the adjacent side, and we are given the value of the hypotenuse. We can set up the ratio for cosine to find the value of angle $x$. The cosine ratio is as follows:
cos $A = \frac{adjacent}{hypotenuse}$
Let's plug in our givens:
cos $x = \frac{3}{4}$
Take $cos^{-1}$ of the fraction:
$x = 41^{\circ}$
