text: geometry question? prove the legs of ans isosceles trapezoid congruent other than by defintion. no, it's not my homework either. sorry. but that's not how it works.  isosceles, from gk. isoskeles "with equal sides," from isos "equal" + skelos "leg".  if the legs aren't congruent, it's not isoceles.  there is nothing to prove.  on the otherhand if the statement of your problem is  that the base angles of the trapazoid are equal, then you prove it like this.   label the corners of of your trapazoid a,b,c,d such that ab, bc, cd, and ad are sides,  ab and cd are the parallel sides, and ab is shorter than cd.  adc and bcd will be equal angles, so will bad and abc.  extend the lines ad and bc until they meet. label the point e.  since adc and bcd are equal, the triangle cde is isoceles as is the triangle abe.  so you have de = ce and ae = be, so ad = de-ae = ce-be = bc.