You can use maths to determine the mitre angles of any handrail system. Just measure one angle, subtract it from 90 degrees and you have your answer. But you don't have the time or materials for trial-and-error cutting on an expensive handrail. Use a mechanical model to ensure accuracy.
Lay your handrail diagram with angles in a large room or garage and draw it on the floor. Draw one angle at a time.
Measure the handrail width. Now cut two pieces of plywood that size, likely 5 to 6.2 cm (2 to 2 1/2 inches) wide by 120 cm (48 inches) long. Lay out the plywood strips, aligning them with your sketched handrail system. Allow one end of the strip to overlap the other end directly above the angle you wish to calculate.
Eyeball the degree of the angle by standing above it, then take a pencil and draw a line through where you think the angle should be. Pick up the strips and take them over to the mitre saw.
Loosen the dial on the mitre saw and swing the blade directly over your mark on the strip. Bring the blade down until it's aligned with the mark. Lock down the dial, turn on the saw and cut the end of both plywood strips at this angle.
Place the strips back down on the floor, aligning them with the handrail drawing. Bring the angles together, measure the gap and divide by half. Re-set the mitre saw by half in the direction needed. Re-cut both pieces. If you still have a gap, repeat until the angle closes and the 120 cm (48 inch) strips are perfectly aligned with your drawing. This is your angle. Write it on the end of the strips, top and bottom. Draw out your next angle on the floor and repeat the steps.
Spin the plywood strips around and use the other ends for the second angle. If you have more than two different angles, cut more strips. If the balusters or "posts" are already in place, lay the plywood strips on top of them, imitating the actual handrail and proceed as above.
Don't throw away the plywood strips. Some mitre angles will be in fractions with no reference setting on the mitre gauge. When you are ready to cut the handrail, lay the plywood strip on the mitre saw table, and swing the blade over to match the pre-cut angle from before.