Congruence and Similarity Tests, Worked Side by Side

Two triangles are congruent when they are identical in every way, same angles and same side lengths, just possibly turned or flipped. They are similar when one is a scaled copy of the other, same shape, same angles, but sides in a fixed ratio. The exam asks you to prove which relationship holds, and the marks come from quoting the correct test, in full, with the correct reason. Naming the test is the question.

The four congruence tests

To prove two triangles congruent you must establish one of these, and you must name it:

SSS, three pairs of equal sides.

SAS, two pairs of equal sides with the equal angle between them.

ASA, two pairs of equal angles with the equal side between them (AAS, the side not between, is the same test in effect).

RHS, in right-angled triangles, the right angle, the hypotenuse, and one other side equal.

Note what is not a test: AAA does not prove congruence, because equal angles only fix the shape, not the size. AAA is a similarity condition, not a congruence one, and confusing the two is the classic lost mark.

The similarity tests

To prove two triangles similar, establish one of these:

AA, two pairs of equal angles (the third is then automatic).

SAS in ratio, two pairs of sides in the same ratio with equal included angles.

SSS in ratio, all three pairs of sides in the same ratio.

In Singapore questions, AA is by far the most common, usually from parallel lines or a shared angle.

Using the scale factor

Once triangles are similar with scale factor k, the ratio runs through three quantities differently, and this is heavily tested:

Lengths scale by k. Areas scale by k2. Volumes scale by k3.

Worked example. Two similar triangles have areas 18 cm^2 and 50 cm^2. The length scale factor is the square root of the area ratio: √(18/50) = 3/5. So if a side of the smaller triangle is 6 cm, the matching side of the larger is 6 x (5/3) = 10 cm. The trap is to scale the length by the area ratio 18/50 directly; you must square-root it first, because length scales by k, not k2.

The teaching point

Congruence is "identical", proved by SSS, SAS, ASA or RHS; similarity is "scaled", proved usually by AA; and AAA never proves congruence. When triangles are similar, length scales by k, area by k2, volume by k3, so you must move between them with the right power. The discipline is to state the test by name and to know which power of the scale factor a question is really asking for.

If your child can see that two triangles "look similar" but cannot write the formal test that earns the mark, that is exactly the gap GPA's Secondary Math programme closes, on real E-Math geometry questions. A short diagnostic consult will show where the reasoning needs tightening.

The scale-factor worked example on this page was checked independently before publishing.