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

Suppose we have two triangles, ABC and PQR. ABC?PQR if AB¯PQ¯, BC¯QR¯, and either BACQPR or BCAQRP.

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

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

Suppose we have a right triangle, XYZ, with right angleMathworldPlanetmathPlanetmath XZY. Let P and Q be two points on XZ equidistant from Z such that P is between X and Z and Q is not. Since XZY is right, this makes PZY right, and P,Q are equidistant from Z, thus 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 YP¯YQ¯. Further, YXP is in fact the same angle as YXQ, thus YXPYXQ. Since XY¯XY¯, XYP and XYQ clearly meet the SSA test, and yet, just as clearly, are not congruent. This results from YXZ being acute. This example also reveals the exception to the ambiguous case, namely XYZ. If R is a point on XZ such that YR¯YZ¯, then RZ. Proving this exception amounts to determining that 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.

Title SSA
Canonical name SSA
Date of creation 2013-03-22 12:28:53
Last modified on 2013-03-22 12:28:53
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 6
Author mathcam (2727)
Entry type Definition
Classification msc 51M99