How do you find an angle measure for the angles XYZ if it looks like it's 63 degrees?

The opposite sides and therefore the opposite angles of a parallelogram are equal and the four angles add up to 360 degrees.

Knowing this and being given one angle of 100 you will know the opposite angle will be 100. Take the sum of the two know angles from 360 and divide by two to find the two remaining angles.

I find doodling a sketch helps with questions like this.
Let x be in this case the supplementary angle.
Supplementary angles add up to 180 degrees.
Thus,

180 = x + (x+124)
180 = 2x + 124
180 - 124 = 2x + 124 -124 (subtract 124 from both sides)
56 = 2x (divide both side by 2 to isolate x)
56/2 = x
28 = x

Other angle is 28 + 124 or 152.

Check 152 + 28 = 180!

We did it correctly!

Good luck,

Paul

Angles may be measured in three angles:degree, radian and gradians (grads).
tan(45 deg)=1
tan(45 radians)=1.61977
If your problems require angles to be measured in degrees, make sure that the calculator's angle unit is set to degrees.
In RUN screen press [SHIFT][MENU] (SetUp) select the angle line and press function key under the tab Deg (should be F1).

Let i be the measure of the angle of incidence; let r be the measure of the angle of refraction. Both angles are measured from the normal to the surface at the incidence point. Let n_i be the index of the first medium where the incident ray lies. Let n_r be the index of refraction of the second medium.
The law of refraction states that n_i*sin(i)=n_r*sin(r)
Assuming that the first transparent medium is air, we have n_i=1. Solving for n_r we find
n_r=(n_i*sin(i))/sin(r)
For i_r=32 degrees, and r=15 degrees
n_r=sin(32)/sin(15)=2.0475

You did not mention the brand or model of your gun, however, you can determine the angle needed by placing a straight edge in line with the path of the driver and measure the angle from this edge to rails.. If the straight edge to the rails measure 90 degrees or makes a right angle, you need "0 degree" nails, if it measures 80 degrees, you need "15 degree" etc. Most generic nials 15 gauge nails will work. No need to by branded nails unless your gun requires an unusual angle. Good luck.

A linear pair of angles is formed when two lines intersect. Two angles are said to be linear if they are adjacent angles formed by two intersecting lines. The measure of a straight angle is 180 degrees, so a linear pair of angles must add up to 180 degrees.

A pair of angles is said to be vertical if the angles share the same vertex and are bounded by the same pair of lines but are opposite to each other. Such angles are congruent and thus have equal measure

It is not clear which angle you are talking about...

In one case the Hypot = 36/(cos(63)) meters. (about 79 meters)

The other case Hypot = 36/(cos(90-63)) meters (about 40 meters)

I'm not sure what a six leg circle is but a circle has 360 degrees, so if you want to divide it into six parts, each triangle has an angle of 60 degrees at the center, and 60 degrees at the other two angles, so each combined angle of the inscribed hexagon is 120 degrees. (That's measured in degree, not reading off a chop saw.)
Hope that helps...