Two angles and a side not between them

Change the position of the side, as we changed the position of the angle before. Take ∆ABC and ∆XYZ with ∠A = ∠X = 35°, ∠C = ∠Z = 75° and BC = YZ = 4 cm.

Here the equal side BC is NOT between the two equal angles — it lies between ∠B and ∠C, while the given angles are at A and C. When the angle moved out from between the sides, SSA failed. Does the same thing happen here?

It does not, and the reason is a fact you already have: the angle sum property. If two angles of a triangle are known, so is the third.

In ∆ABC, ∠B + 35° + 75° = 180°, so ∠B + 110° = 180°, giving ∠B = 70°. In ∆XYZ the same arithmetic gives ∠Y = 70°. So ∠B = ∠Y, and we did not have to measure it.

Now look again. We have ∠B = ∠Y, BC = YZ and ∠C = ∠Z — two angles with the side BETWEEN them. That is exactly the ASA condition, which we know guarantees congruence.

So ∆ABC ≅ ∆XYZ. The case is called the AAS — Angle Angle Side — condition, and as we have just shown, AAS does guarantee congruence.

Notice the shape of that argument. We did not build a new rule from scratch; we turned the new case into one we had already settled. That is how most of geometry grows, and it is why the angle sum property was worth proving in Part 1.