Right triangles, and the five conditions

SSA does not guarantee congruence in general. But there are special cases where it does, and one of them is important enough to have its own name.

Take ∆ABC and ∆XYZ, both right-angled, with ∠B = ∠Y = 90°, BC = YZ = 4 cm and AC = XZ = 5 cm. Notice that the equal angle is not between the equal sides — so this is an SSA arrangement. Are they congruent?

Construct and find out. Step 1: draw the base QR of length 4 cm. Step 2: draw a line l perpendicular to QR at Q. Step 3: from R, cut an arc on l of radius 5 cm. Step 4: let P be where the arc meets l, and join PR.

Now the worry from the SSA case: does the arc meet the line twice? Extend l downwards below QR and it does — the arc reaches that extension too, giving a second triangle.

But look at that second triangle. The perpendicular at Q is a line of symmetry for the whole picture: everything below QR is the mirror image of everything above it. So the second triangle is congruent to ∆PQR, not different from it — exactly as in the SSS case, and unlike the general SSA case.

So all triangles with these measurements are congruent, and ∆ABC ≅ ∆XYZ. The parts that matched were the right angle, and two other sides, one of which is opposite the right angle. That side has a name: the HYPOTENUSE — always the longest side of a right triangle, the one facing the right angle.

This is the RHS — Right Hypotenuse Side — condition. So the full list of conditions sufficient to guarantee congruence is five:

• SSS — three sides
• SAS — two sides and the angle between them
• ASA — two angles and the side between them
• AAS — two angles and a side not between them
• RHS — right angle, hypotenuse and one side

AAA and general SSA are not among them.