In ∇QWE, m∠Q=45 and m∠E=90. If EW=7, find QE and QW

My response will be based on the assumption that you mean triangle QWE rather than DeltaQWE and also that we are working in a plane.

Since the interior angles sum to 180, that means angle W=45 and we have an isosceles right triangle.  With that, knowing one leg, EW=7, guarantees the other leg is congruent so WE=7 as well. 

In a 45-45-90 triangle the hypotenuse is always longer than the legs by a factor of sqrt(2), ergo QW=7sqrt(2)

Yes I mean triangle there just wasnt a thing for it :P