Congruent and similar are not the same word
Similar figures — Two polygons are similar if their corresponding angles are equal and their corresponding sides are in the same ratio. For triangles, either condition on its own is enough to force the other.
Two figures are congruent if one can be placed exactly on top of the other: same shape and same size. They are similar if they have the same shape but not necessarily the same size — an enlargement or reduction of one another.
Congruence is the special case of similarity where the scale factor happens to be 1. Every congruent pair is similar; almost no similar pair is congruent.
| Test | Applies to | Meaning |
|---|---|---|
| AAA (or AA) | triangles | two pairs of equal angles force the third, so the triangles are similar |
| SSS | triangles | all three pairs of sides in the same ratio |
| SAS | triangles | two pairs of sides in ratio with the included angles equal |
| Angles + ratios | other polygons | both conditions must be checked — equal angles alone is not enough |
Equal angles is not enough outside triangles
A square and a 2 cm × 5 cm rectangle both have four right angles, but they are not similar. Triangles are special: fixing the angles fixes the shape, which is why AA works for them and only for them.
The scale factor, and the two powers of it
If every length in a figure is multiplied by k, then every area is multiplied by k² and every volume by k³. This is not a separate rule to memorise; it falls straight out of the fact that area is two lengths multiplied and volume is three.
A 3 cm cube has surface area 54 cm² and volume 27 cm³. Double every edge and you get a 6 cm cube with surface area 216 cm² (four times as much) and volume 216 cm³ (eight times as much). Lengths ×2, areas ×4, volumes ×8.
Push k to 2. The length bar doubles, the area bar quadruples and the volume bar grows eightfold. This is why a model at 1:10 scale needs a thousandth of the material, not a tenth of it.
Working with corresponding parts
Almost every mark lost in this chapter comes from pairing the wrong sides. Before writing any ratio, name the triangles so that corresponding vertices appear in corresponding positions: writing "triangle ABC is similar to triangle PQR" states that A matches P, B matches Q and C matches R.
Once the order is fixed, the ratios follow mechanically: AB/PQ = BC/QR = CA/RP.
In triangles ABC and PQR, ∠A = ∠P, ∠B = ∠Q, AB = 6 cm, BC = 9 cm and PQ = 4 cm. Find QR, and the ratio of the areas.
- Two pairs of angles are equal, so by AA the triangles are similar with A↔P, B↔Q, C↔R.The third pair of angles is then equal automatically, since angles in a triangle sum to 180°.
- Scale factor from ABC to PQR:
k = PQ/AB = 4/6 = 2/3.Use the pair of sides you know completely. Keep the direction consistent for the rest of the question. QR = k × BC = (2/3) × 9 = 6cm.QR corresponds to BC because Q↔B and R↔C.- Area ratio
= k² = (2/3)² = 4/9.Areas scale by the square, so triangle PQR has 4/9 of the area of ABC.
QR = 6 cm; area of PQR : area of ABC = 4 : 9
The intercept theorem
A line drawn parallel to one side of a triangle cuts the other two sides in the same ratio. This is really the AA similarity test wearing a disguise: the small triangle at the top and the whole triangle share an angle and have equal corresponding angles at the parallel line, so they are similar.
It is the result behind most "find x" diagrams in this chapter, and it works in both directions — if the two sides are cut in the same ratio, the line must be parallel.
Which ratio does the question want?
The two forms above are both correct but they are not interchangeable. AD/DB compares the two pieces of a side; AD/AB compares a piece with the whole side. Only the second form can be used with DE/BC, because DE corresponds to the whole of BC. Decide which one you need before substituting numbers.
Before you leave this chapter
- Similar = same shape, corresponding angles equal and corresponding sides in a fixed ratio k.
- For triangles, AA is enough. For any other polygon you must check angles and ratios.
- Lengths ×k, areas ×k², volumes ×k³. From an area ratio, square-root to recover k.
- Name the triangles in corresponding order before writing a single ratio.
- A line parallel to one side of a triangle divides the other two sides in the same ratio.
Congruence: the four tests, and the one that is not a test
Similarity asks whether two figures have the same shape. Congruence asks whether they are identical, and it has its own set of tests that the paper expects by name.
For triangles, any one of these four is enough to prove congruence: SSS (three sides), SAS (two sides and the angle between them), ASA (two angles and the side between them, or AAS which reduces to it), and RHS (right angle, hypotenuse and one other side).
What is not a test is AAA. Three matching angles prove similarity, never congruence — any two equilateral triangles have identical angles and can be any size at all. The other near-miss is SSA, two sides and a non-included angle, which can produce two genuinely different triangles from the same data.
Why SAS insists on the included angle
Fix two sticks of length 5 cm and 7 cm hinged at a point. The angle at the hinge decides the third side completely, so SAS pins the triangle down. Give the 5 cm side, the 7 cm side and an angle not between them and the third vertex can often be placed in two different positions — which is exactly why SSA is rejected as a test.