Use your compass to find the length of each right triangle's hypotenuse, the longest side of the triangle. If the hypotenuses are different lengths, set your compass to the length of the shorter one. Otherwise, set it to the length of either.
Place the tip of your compass at one end of the hypotenuse of the larger triangle. Mark that point Q. Make a small arc that intersects that hypotenuse. The distance between the arc and the point where you placed the tip of your compass should equal the hypotenuse of the smaller triangle. Mark as P the point where the arc intersects the hypotenuse.
Set your compass to the length of the second-longest side of the smaller triangle. Place the tip of your compass at point Q and draw a complete circle.
Set your compass to the length of the smallest side of the smaller triangle. Find the first are you drew and place the tip of your compass at point P, where that arc intersects the hypotenuse of the larger triangle. Draw another complete circle.
Find the two points where the two circles intersect. One point will lie inside the larger triangle, or outside it so that one the larger triangle's sides is between the point and the hypotenuse. The second point will lie outside the larger triangle so that the hypotenuse is the side closest to that point. Mark that point N.
Use your straightedge to draw line segments from P to N and from Q to N. If the hypotenuse of each triangle is the same length, you are finished. If not, extend line segment QN to the edge of the paper. Label the remaining end of the larger triangle's hypotenuse R. Construct a line segment passing through R and parallel to line segment PN. Extend this line segment until it intersects QN.
Repeat the process with isosceles right triangles, equilateral triangles and similar non-isosceles right triangles. Determine which yields a non-square rectangle, which yields a square and which yields a non-rectangular parallelogram.