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, and . if , , and either or .
Since this method does not prove congruence, it is more useful for disproving it. If the SSA method is attempted between and and fails for every ,, and against every ,, and , then .
Suppose and the SSA test. The specific case where SSA fails, known as the ambiguous case, occurs if the congruent angles, and , are acute. Let us illustrate this.
Suppose we have a right triangle, , with right angle . Let and be two points on equidistant from such that is between and and is not. Since is right, this makes right, and , are equidistant from , thus bisects and , and as such, every point on that line is equidistant from and . From this, we know is equidistant from and , thus . Further, is in fact the same angle as , thus . Since , and clearly meet the SSA test, and yet, just as clearly, are not congruent. This results from being acute. This example also reveals the exception to the ambiguous case, namely . If is a point on such that , then . Proving this exception amounts to determining that 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.
|Date of creation||2013-03-22 12:28:53|
|Last modified on||2013-03-22 12:28:53|
|Last modified by||mathcam (2727)|