I am trying to find the angles that allows this polygon to fully connect. the line segments are to stay the same length and the two right angles need to be right angles. i have looked for calculators and formulas that would help find the remaining three angles.

Construct your two rt. angles with the appropriate dimensions. At the end of each line draw an arc (or circle) whose radius matches the desired line length and the center is at the end of the line. Where the two circles intersect is your vertex.