# Does SSA congruence criterion work if the non-included angle is obtuse?

I was trying to make some triangles when $$2$$ sides and the non-included angle was given. When the angle given was below $$90$$ degrees, there were two possible triangles which could be made. But, when it went above $$90$$ degrees, only one possibility came. I was using a compass to determine the third side's possibilities. I checked the internet if there were questions like this, but there were none. Also, no site said specifically that it works for $$90+$$ angles.
Does this mean that the SSA congruence can work for angles greater than $$90$$ degrees? If so, I was taught very wrong in school.

• See SSA : "If two triangles satisfy the SSA condition and the length of the side opposite the angle is greater than or equal to the length of the adjacent side (SSA, or long side-short side-angle), then the two triangles are congruent. The opposite side is sometimes longer when the corresponding angles are acute, but it is always longer when the corresponding angles are right or obtuse." – Mauro ALLEGRANZA Feb 21 at 11:01
• What I call "Side-Side-non-acute-Angle" (SSnaA) does work, as you have determined. That said, it's not fair to say that you were taught "very wrong" in school. "Side-Side-Angle" is not a valid congruence pattern, due to the ambiguity in the acute case; the nature of the angle is additional information that sometimes resolves any ambiguity, but that information isn't always available. (Compare SAS or ASA, which work without additional information.) Note: If you were taught the "Hypotenuse-Leg" congruence pattern, then you were taught "Side-Side-right-Angle", so there's that. :) – Blue Feb 21 at 13:51

## 1 Answer

There is the following theorem.

Let $$AB=PQ$$, $$BC=QR$$, $$\measuredangle A=\measuredangle P$$ and $$BC>AB$$.

Thus, $$\Delta ABC\cong\Delta PQR.$$

If $$\measuredangle A\geq 90^{\circ}$$ we can use this theorem immediately, but it's true also for $$\measuredangle A<90^{\circ}.$$