Writing congruence properly

Two triangles are congruent — but HOW should they be laid on top of each other? Which vertices overlap? The answer must put the EQUAL sides together.

For a pair ∆ABC and ∆XYZ, suppose A must go over X, B over Y and C over Z for the equal sides to meet. Then when they overlap exactly, their angles match too: ∠A = ∠X, ∠B = ∠Y and ∠C = ∠Z.

So congruent triangles have CORRESPONDING vertices (A and X, B and Y, C and Z), corresponding SIDES (AB and XY, BC and YZ, AC and XZ) and corresponding ANGLES.

The notation carries all that information. We write ∆ABC ≅ ∆XYZ, and the ORDER of the letters is doing real work: the first vertex named in ∆ABC corresponds to the first named in ∆XYZ, the second to the second, the third to the third.

So for this pair it would be WRONG to write ∆ACB ≅ ∆XYZ, which claims C corresponds to Y. But ∆ACB ≅ ∆XZY is perfectly correct, because it keeps the same pairings in a different order.

Try it on a rectangle ABCD, split by the diagonal BD into ∆ABD and ∆CDB. Since ABCD is a rectangle, AB = CD and AD = CB. The remaining side, BD, is COMMON to both triangles. So all three sides match and the SSS condition gives congruence.

Now which vertices correspond? Suppose we guessed A with C, B with B, D with D. That correspondence lays side AB over side CB — and those need not be equal, so the superimposition fails. Work out the right correspondence instead, and express the congruence with the letters in that order.