Congruency of triangles: is SSA rule invalid?

In the previous page we have known that if two or more figures have exactly the same shape and size, then they are known to be congruent. We have also seen that the five rules, namely SSS, SAS, ASA, AAS and RHS can be applied to analyze the congruency of two or more triangles.

To repeat our findings on triangles, there are mainly six parts in a triangle, three sides and three angles, where three angles always sum up to 180°, and the sum of any two sides is always greater than the third side.

For any \(▲ABC\), there are three angles \(∠A, ∠B\) and \(∠C\). Side \(AB\) is opposite to \(∠C\), and also reffered to as \(c\). Similarly side \(BC\) is opposite to \(∠A\) and reffered to as \(a\), side \(CA\) is opposite to \(∠B\), and reffered to as \(b\).

To draw a unique triangle, we require minimum three inputs, including one side of the triangle to be drawn. You may kindly refer to the page Triangle Calculator where you can get any triangle drawn by providing suitable input data like three sides(SSS), two sides and the angle inbetween (SAS) etc.

When you provide two sides and a non-included angle as input, you may get one, or two or no triangle. Try to draw a triangle with the dataset \(c = 13, a = 9\) and \(∠A = 30°\); you will end up with two different triangles, of course which are not congruent. The following animation attempts to analyze what is happening in such situation.

Why Side-Side-Angle(SSA) is not a congruency rule?

SSA criteria of triangle congruency is not valid, and sometimes called ambiguous, because with given two sides and a non-included angle sometimes two different triangles can be drawn.

Let us take the example of a triangle \(ABC\) whose \(c = 60, a = 32\), and \(∠A = 30°\).

In the above the two sides \(c = 60, \,a = 32\) and one non-included angle \(∠A = 30°\) are fixed by input.
Draw a circle taking vertex \(B\) as the center and side \(a\) as the radius. We find that the circle intersects side \(b\) at two points \(C_1\) and \(C_2\) because \(b > \perp BC_3\). Due to this, two triangles can be formed; \(▲ABC_1\) where side \(a = BC_1\), and \(▲ABC_2\) where side \(a = BC_2\).

We have drawn an additional segment \(BC_3 \perp CA\) which is the distance between vertex \(B\) and side \(b\), \(BC_3=30\). So, \(BC_3\) is the minimum distance between vertex \(B\) and side \(b\).

• ● If side \(a \lt BC_3\), no triangle is possible
• ● if side \(a = BC_3\), only one right-angled triangle is possible
• ● if side \(a \gt BC_3\) and \(a \lt c\), two triangles are possible (as above)
• ● if side \(a \gt c\), only one triangle is possible

If we take SSA rule to be valid, then the above two triangles should have been congruent, because \(c=60, a=32, ∠A=30°\), and \(f=60, d=32, ∠D=30°\), but actually they are not congruent.This is why the SSA criteria should not be applied in detemining the congruency of triangles.

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