In most geometry courses, we learn that there's no such thing as "SSA Congruence". That is, if we have triangles ABC and DEF such that AB = DE, BC = EF, and angle A = angle D, then we cannot deduce that ABC and DEF are congruent.

Question

However, there are a few special cases in which SSA "works". That is, suppose we have AB = DE = x, BC = EF = y, and angle A = angle D = theta. For some values of x, y, and theta, we can deduce that triangle ABC is congruent to triangle DEF. Use the Law of Cosines or Law of Sines to explain the conditions x, y, and/or theta must satisfy in order for us to be able to deduce that triangle ABC is congruent to triangle DEF. (In other words, find conditions on x, y, and theta, so that given these values, you can uniquely reconstruct triangle ABC.)

I know that one case is if theta is 90 degrees.

Answer 1

In this case, the Law of Cosines can be used to determine that x^2 + y^2 = (x+y)^2.

In the case where theta is not 90 degrees, the Law of Sines can be used to determine that (x/sin A) = (y/sin D).

Therefore, for SSA Congruence to work, x, y, and theta must satisfy either x^2 + y^2 = (x+y)^2 or (x/sin A) = (y/sin D).