# Calculating angles when sides are known - Without Trignometric ratios.

The question is:

ABCD is a parallelogram and BFDE is a square . If AB is 20 and CF is 16 what is the perimeter of the parallelogram.

The question is fairly simple and I know how to solve it. However how would I get the remaining angles of triangle FDC (highlighted) if i know that CD=20 and FD=12. I also know that angle F is 90 however the triangle doesn't seem to be 30-60-90 or a 45-45-90. Also I need to solve this without Trigonometric ratios since I wont be using calculator

-
Your question is basically "What are the angles of a 3-4-5" triangle, which is pretty googlable. The short answer is that you need to use trig functions to express these numbers. – Cam McLeman Jun 19 '12 at 0:08
I realize that but I could obtain these angles using trigonometric ratios. I was just curious if it was possible to obtain them without trigonometric ratios which i guess is not possible. – MistyD Jun 19 '12 at 0:12

The triangles $\,ABE\,,\,CDF\,$are congruent, thus $\,CD=20\,$. Apply now Pythagoras and get $\,FD^2=20^2-16^2=144\Longrightarrow FD=12=FB=ED\,$ , so $\,AD=BC=16+12=32\,,\,AB=CD=20\,$ and now just add.
The angle is not at all pleasant. In particular (in radians) it is not a rational multiple of $\pi$. – André Nicolas Jun 19 '12 at 0:55