We have an isosceles triangle, what is the theorem called that states that the sides opposite it's congruent angles will have congruent lengths? Could someone also explain why this is.
The theorem you want is:
If in a triangle two angles equal one another, then the sides opposite the equal angles also equal one another.
This works for any triangle, not just an isoceles one (obviously, the theorem implies that the triangle in question is isoceles, but you don't need to know that in advance).
This is an early theorem in Euclid's Elements. It's a simple consequence of the converse, that if a triangle has two equal sides then the opposite angles must also be equal, which is also proven in Euclid's elements.