SSA Congruence: Constraints

In my last post, I pointed out that SSA is in fact sufficient for determining all three sides and angles under certain conditions. In this post, I will specify those conditions, with illustrations.

Given two noncollinear segments \(\overline{S_1}\) and \(\overline{S_2}\) and angle \(\angle A\), where  \(\overline{S_1}\)’s two endpoints are the vertex of \(\angle A\) and an endpoint of \(\overline{S_2}\), we can create zero, one, or two triangles, depending on the various measurements.

Here’s a case where we can create two triangles:

SSA-Acute-1 SSA-Acute-2

In both cases of \(\triangle ABC\), \(m\angle A \approx 38^\circ\), \(AB = 2\), and \(BC = 1.4\). I’ve included the circle to show the source of the ambiguity and why it only holds in some cases. If the radius of \(\odot B\) is too small to reach \(\overrightarrow{AC}\), then no triangles can be formed; if it’s just long enough, then \(\triangle ABC\) will be a right triangle with \(\angle C = 90^\circ\). If it’s any longer than that, and \(\angle A\) is acute, then \(\odot B\) will intersect \(\overrightarrow{AC}\) in two places, each of which represents the vertex of a valid triangle.

What if \(\angle A\) is obtuse? Why won’t \(\odot B\) intersect \(\overrightarrow{AC}\) twice then? Here is that condition illustrated:


Notice that while \(\odot B\) intersects \(\overleftrightarrow{AC}\) twice, it only intersects \(\overrightarrow{AC}\) once. Also notice that if \(\angle A\) is right or obtuse, then \(BC\) (that is, \(S_2\)) has to be longer than \(AB\) (that is, \(S_1\)).

Returning to the case that \(\angle A\) is acute: Under what circumstance will \(\odot B\) intersect \(\overrightarrow{AC}\) exactly once? There are actually two cases where this is true.

\(\odot B\) intersects \(\overrightarrow{AC}\) exactly once when \(\angle A\) is acute and \(\angle C\) is right. In the left diagram, \(AC = 0.5\); in the right, \(AC = 1.2\).


\(\triangle ABC\) has \(\angle C = 90^\circ\) when \(\sin{\angle A} = \frac{S_2}{S_1}\), that is, when \(S_2 = S_1 \sin{\angle A}\).

Also, if \(S_1 = S_2\), then we have an isosceles triangle, and \(\odot B\) intersects \(\overrightarrow{AC}\) exactly once (more rigorously stated, \(\odot B\) intersects \(\overrightarrow{AC}\) twice, but one of these is at \(A\)).

SSA-Isosceles-1 SSA-Isosceles-2

If \(S_2 > S_1\), then \(\odot B\)’s second intersection with \(\overleftrightarrow{AC}\) is not on \(\overrightarrow{AC}\):


When \(\angle A\) is acute, here are the possibilities:

Case Triangles
\(S_2 < S_1 \sin{A}\) 0
\(S_2 = S_1 \sin{A}\) 1
\(S_1 \sin{A} < S_2 < S_1 \) 2
\(S_1 \le S_2 \) 1

When \(\angle A\) is right or obtuse, then \(S_2\) must be greater than \(S_1\); if it is, then a unique triangle will be formed. This gives us a complete list of conditions:

Angle type of \(\angle A\) Length of \(S_2\) Triangles
Acute \(S_2 < S_1 \sin{A}\) 0
Not acute \(S_2 \le S_1 \) 0
Acute \(S_2 = S_1 \sin{A}\) 1
Acute \(S_1 \le S_2 \) 1
Not acute \(S_1 < S_2 \) 1
Acute \(S_1 \sin{A} < S_2 < S_1 \) 2

In conclusion, then, SSA usually works. In some cases, \(S_2\) is too short to create a valid triangle because it simply won’t reach the ray created by \(S_1\) and \(\angle A\). In other cases, when \(\angle A\) is acute and \(S_2\) is between \(S_1 \sin{A}\) and \(S_1\), there are two possible triangles. In all other cases, SSA will create a unique triangle.

(Images created in GeoGebra.)

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