SSA is a method for determining whether two triangles are congruent by comparing two sides and a non-inclusive angle. However, unlike SAS, SSS, ASA, and SAA, this does not prove congruence in all cases.

Suppose we have two triangles, $\triangle ABC$ and $\triangle PQR$ . $\triangle ABC \cong? \triangle PQR$ if $\overline{AB} \cong \overline{PQ}$ , $\overline{BC} \cong \overline{QR}$ , and either $\angle BAC \cong \angle QPR$ or $\angle BCA \cong \angle QRP$ .

Since this method does not prove congruence, it is more useful for disproving it. If the SSA method is attempted between $\triangle ABC$ and $\triangle PQR$ and fails for every $ABC$ ,$BCA$ , and $CBA$ against every $PQR$ ,$QRP$ , and $RPQ$ , then $\triangle ABC \not\cong \triangle PQR$ .

Suppose $\triangle ABC$ and $\triangle PQR$ meet the SSA test. The specific case where SSA fails, known as the ambiguous case, occurs if the congruent angles, $\angle BAC$ and $\angle QPR$ , are acute. Let us illustrate this.

Suppose we have a right triangle, $\triangle XYZ$ , with right angle $\angle XZY$ . Let $P$ and $Q$ be two points on $\overleftrightarrow{XZ}$ equidistant from $Z$ such that $P$ is between $X$ and $Z$ and $Q$ is not. Since $\angle XZY$ is right, this makes $\angle PZY$ right, and $P$ ,$Q$ are equidistant from $Z$ , thus $\overleftrightarrow {YZ}$ bisects $P$ and $Q$ , and as such, every point on that line is equidistant from $P$ and $Q$ . From this, we know $Y$ is equidistant from $P$ and $Q$ , thus $\overline{YP} \cong \overline{YQ}$ . Further, $\angle YXP$ is in fact the same angle as $\angle YXQ$ , thus $\angle YXP \cong \angle YXQ$ . Since $\overline{XY} \cong \overline{XY}$ , $\triangle XYP$ and $\triangle XYQ$ clearly meet the SSA test, and yet, just as clearly, are not congruent. This results from $\angle YXZ$ being acute. This example also reveals the exception to the ambiguous case, namely $\triangle XYZ$ . If $R$ is a point on $\overleftrightarrow{XZ}$ such that $\overline{YR} \cong \overline{YZ}$ , then $R \cong Z$ . Proving this exception amounts to determining that $\angle XZY$ is right, in which case the congruency could be proven instead with SAA.

However, if the congruent angles are not acute, i.e., they are either right or obtuse, then SSA is definitive.