In math class, a bonus homework question asked, “Given SSA, what are the necessary and sufficient conditions to prove that the triangle will be acute?”

In other words, create a test for SSA (side-side-angle) that a triangle will pass if it is acute.

My proposition: Given SSA, it must be an isosceles triangle (S1 = S2) with a given angle > 45.

Proof (by exhaustion):

Let S1 = Adjacent side (w.r.t to A) and S2 = Opposite side.

If adj*Sin(A) > opposite, obviously there will be no resulting triangle.

If adj*Sin(A) < opp, but opp > adj, there will only be one, obtuse triangle.

If adj*Sin(A) < opp, and opp < adj, there will be TWO resulting triangles; since both of them are not acute, we cannot *assume* that, given SSA in this situation, the triangle is acute.

This leaves us with adj*Sin(A) < opp and **opp = adj** (that is, an isosceles triangle).

Since we also assume A > 45 and that the triangle is isosceles, we know that the angle on the bottom of the opposite leg is equal, we know that 2A > 90, and thus the last angle is acute. Therefore, it’s an acute triangle.

But I’m getting reports that this wasn’t the answer. Can anyone find where I went wrong?