Answer
$x=4,\ z=4\sqrt{2},\ y=4\sqrt{3},\ w=8$
Work Step by Step
We see two right triangles, one to the left, one to the right.
The triangle to the left (x-4-z), is a $45^{\mathrm{o}}-45^{\mathrm{o}}$ right triangle,
which is isosceles,
so $x=4.$
In a $45^{\mathrm{o}}-45^{\mathrm{o}}$ right triangle, the hypotenuse is $\sqrt{2}$ times the measure of a leg.
So, $z=4\sqrt{2}$
The triangle to the right (4-y-w) is a $30^{\mathrm{o}}-60^{\mathrm{o}}$ right triangle
In the $30^{\mathrm{o}}-60^{\mathrm{o}}$ right triangle, the side opposite the $60^{\mathrm{o}}$ angle is $\sqrt{3}$ times as long as the side opposite to the $30^{\mathrm{o}}$ angle.
So, $y=4\sqrt{3}$
In the $30^{\mathrm{o}}-60^{\mathrm{o}}$ right triangle, the length of the hypotenuse is 2 times as long as the shorter leg (opposite the $30^{\mathrm{o}}$ angle).
So, $w=2(4)=8$
