Two shapes are congruent if they are identical in size and shape — one can be placed exactly on top of the other (possibly after flipping or rotating). Two shapes are similar if they have the same shape but a different size — all corresponding angles are equal and all corresponding sides are in the same ratio (the scale factor).
What does congruent mean?
Congruent shapes are exactly the same shape and the same size. If you were to cut them out of paper, one would fit perfectly over the other — even if you need to rotate or reflect it first.
Key point: congruent shapes can be rotated, reflected, or translated — these transformations do not change size or shape. However, enlargement changes size, so an enlarged shape is similar but not congruent.
Examples of congruent shapes:
- Two 5 cm × 3 cm rectangles.
- The two halves of a regular hexagon cut along a line of symmetry.
- Any two equilateral triangles with the same side length.
What are the four conditions for congruent triangles?
At KS3 and beyond, two triangles are congruent if any one of these four conditions holds:
| Condition | What it means |
|---|---|
| SSS | All three sides are equal |
| SAS | Two sides and the included angle (between them) are equal |
| ASA | Two angles and the included side (between them) are equal |
| RHS | Right angle, hypotenuse, and one other side are equal |
Worked example — SSS: Triangle ABC has sides 5 cm, 7 cm, 9 cm. Triangle DEF has sides 9 cm, 5 cm, 7 cm. Are they congruent?
Yes — both have the same three side lengths (SSS), so they are congruent.
Note: AAA (three equal angles) is NOT enough for congruence — it only tells you the triangles are similar.
What does similar mean?
Two shapes are similar if they have exactly the same shape but are different sizes. For similar shapes:
- All corresponding angles are equal.
- All corresponding sides are in the same ratio (the scale factor).
Enlargement always produces a similar shape. Every circle is similar to every other circle. Squares are all similar to each other.
What is the scale factor?
The scale factor (SF) tells you how much larger or smaller the image is compared to the original.
Scale factor = length on image ÷ corresponding length on original
| Original length | Image length | Scale factor |
|---|---|---|
| 4 cm | 6 cm | 6 ÷ 4 = 1.5 |
| 10 cm | 4 cm | 4 ÷ 10 = 0.4 |
| 3 cm | 9 cm | 9 ÷ 3 = 3 |
If the scale factor is greater than 1, the shape is enlarged. If it is between 0 and 1, the shape is reduced.
How do you find a missing length using similarity?
Worked example: Triangles PQR and XYZ are similar. PQ = 6 cm, QR = 8 cm, PR = 10 cm. XY = 9 cm. Find YZ and XZ.
- Find the scale factor: SF = XY ÷ PQ = 9 ÷ 6 = 1.5.
- YZ = QR × SF = 8 × 1.5 = 12 cm.
- XZ = PR × SF = 10 × 1.5 = 15 cm.
Always identify which sides correspond to each other — match the sides opposite equal angles.
What are the key differences between congruence and similarity?
| Feature | Congruent | Similar |
|---|---|---|
| Same shape | Yes | Yes |
| Same size | Yes | Not necessarily |
| Angles | All equal | All equal |
| Sides | All equal | Proportional (same ratio) |
| Scale factor | Always 1 | Can be any positive value |
Frequently asked questions
Can a shape be congruent to itself?
Yes — any shape is congruent to itself. This is called the identity congruence. In geometry proofs, this trivial case is sometimes used to anchor arguments, but in KS3 questions you will be comparing two distinct shapes.
Is a reflection congruent to the original?
Yes — reflection, rotation, and translation all produce congruent images. These are called isometric (same-measure) transformations because they preserve both length and angle. Enlargement is not isometric because it changes length.
How are similar shapes used in real life?
Maps and scale drawings use similarity — a map is a similar image of the real landscape. Architects produce scale drawings that are similar to the buildings they represent. Photographs are similar to the scenes they capture. Shadows and sunbeams also create similar triangles, which is how the ancient Greeks estimated the height of the pyramids.
How do I know which sides are corresponding in two similar triangles?
Corresponding sides are opposite equal angles. If angle A = angle X, angle B = angle Y, angle C = angle Z, then side BC (opposite A) corresponds to side YZ (opposite X). Label angles first, then match the sides.
For Socratic KS3 geometry practice on congruence and similarity, see aitutors.me.