SSA Congruence for Obtuse Triangles

In most Geometry classes, students are taught five different ways to prove two triangles congruent: SSS, SAS, ASA, AAS, and HL.  Each “S” stands for “side”, and each “A” stands for “angle”, so SSS means that two triangles can be proven congruent if all three sides are congruent, SAS means that two triangles can be proven congruent if two sides and the included angle are congruent, ASA means that two triangles can be proven congruent if two angles and the included side are congruent, and AAS means that two triangles can be proven congruent if two angles and the non-included side are congruent.  The last congruence, HL, stands for “hypotenuse-leg”, which means that two right angle triangles can be proven congruent if one leg and the hypotenuse are congruent.

Traditional Triangle Congruence Theorems

Students are also taught that SSA (or ASS) is not a way to prove triangles congruent, because in some cases two different triangles can be formed by two given sides and a non-included angle.  With a picture of this counter-example and a predictable lame joke about how we should have no postulates named after a donkey, the teacher (or text) closes the discussion on the matter.

SSA Counter-Example

However, SSA is a legitimate congruence theorem if the given angle is not an acute angle.  This is already evident for right angle triangles, because the HL congruence theorem is really an SSA congruence theorem in disguise.  But it is also true when the given angle is obtuse as well.  In the typical counter-example for SSA, the law of sines and the identity sin(x) = sin(180° – x) can be used to prove that the two different (non-right) triangles have a second angle that is always acute in one case and always obtuse in the other case:

01	by the law of sines, sin B = b/a sin A
02	by a trigonometric identity, sin x = sin (180° – x)
03	therefore, two possible solutions for ∠B are x and 180° – x
04	in a non-right triangle, x is either acute or obtuse
05	if x is acute, then 180° – x is obtuse
06	if x is obtuse, then 180° – x is acute
07	either way, one solution for ∠B is acute and one solution for ∠B is obtuse

However, if the given first angle is already obtuse, then there cannot be a second obtuse angle because the angle sum of the triangle would be more than 180°.  This leaves only one possible way to form the triangle, which means SSA is a legitimate congruence theorem when the given angle is obtuse.

SSA Congruence

(if the given angle is obtuse)

                                                                                  

Most mainstream Geometry texts only teach five different ways to prove two triangles congruent (SSS, SAS, ASA, AAS, and HL), and only mention SSA as an incorrect way to prove two triangles congruent.  Indeed, SSA is incorrect if the given angle is acute, because the given criteria can produce two different possible triangles.  However, if the given angle is right or obtuse, only one triangle can be produced, in which case SSA is a legitimate congruence theorem.