# 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:

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 $$A$$).

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.)