Three sides are enough

Meera and Rabia have to make a cardboard cutout identical to a triangular frame at school. The frame is too big to be traced on paper, so they must measure it instead.

With a measuring tape they find the sides to be 40 cm, 60 cm and 80 cm. Rabia then takes out her protractor to measure the angles — and Meera stops her. 'The angles are not required! With the side lengths we have measured, we can create a triangle congruent to this one.'

Do you agree with Meera? Scale the problem down to 4 cm, 6 cm and 8 cm so it fits on a page, and try it.

Rabia raises a real objection. 'If I were to construct this triangle, I would draw a segment of one of the lengths, say AB = 6 cm, and then draw circles from its end points with radii 4 cm and 8 cm. But the circles intersect at TWO points, forming two triangles, ∆ABE and ∆ABF. Do these have the same shape and size? If not, we will not know which one is congruent to the original.'

Good objection — so examine the two triangles. Trace one and compare; or take a cutout and superimpose it; or notice that AB acts as a line of symmetry, because the construction above the line and below it was exactly the same act.

By any of those methods, ∆ABE and ∆ABF turn out to be congruent. And since the construction was completely general, ALL triangles with the same three side lengths are congruent. Rabia's two triangles were never a problem — they were two copies of the same triangle.

So Meera was right: the side lengths are sufficient. If two triangles have the same side lengths, then they are congruent. This is the SSS — Side Side Side — condition for congruence. The order in which you read the sides does not matter: a triangle with sides 4, 6, 8 and one with sides 8, 4, 6 have the same three lengths, so they are congruent.