Two ideas, side by side
It helps to keep the two words apart from the start. Congruent shapes match exactly: same angles and same lengths, just possibly turned or flipped. Similar shapes have the same angles but their lengths are all multiplied by a fixed number, so one is an enlargement of the other.
- Congruent
- The same shape and the same size. One can be placed exactly on top of the other after rotating, reflecting, or sliding it.
- Similar
- The same shape but a different size. Corresponding angles are equal and corresponding sides are in the same ratio.
Congruence conditions
You do not need to check every side and every angle to prove two triangles are congruent. Any one of these four sets of matching parts is enough:
| Condition | What must match |
|---|---|
| SSS | All three pairs of sides. |
| SAS | Two pairs of sides and the angle between them (the included angle). |
| ASA | Two pairs of angles and the side between them. |
| RHS | In right-angled triangles: the right angle, the hypotenuse, and one other side. |
The angle must be included for SAS
SAS works only when the known angle sits between the two known sides. Two sides and a non-included angle (sometimes written SSA) is not a valid condition, it can give two different triangles.
Similar triangles and scale factor
Two triangles are similar when their corresponding angles are equal. Once that is true, every pair of corresponding sides shares the same ratio, and that ratio is the scale factor.
To spot which sides correspond, match them by the equal angles they sit opposite. The longest side of one triangle corresponds to the longest side of the other.
- Scale factor
- The number every length is multiplied by to go from one similar shape to the other. A scale factor greater than 1 enlarges; between 0 and 1 it shrinks.
Finding a missing side
Once you know two shapes are similar and you have one matching pair of sides, you can find the scale factor and use it on any other side.
Triangle \(ABC\) is similar to triangle \(DEF\). In \(ABC\), \(AB = 4\) cm and \(BC = 6\) cm. The matching side \(DE = 6\) cm. Find \(EF\).
Where this is assessed
Naming the correct congruence condition and justifying each matched part is prime Criterion C (Communicating) work: a proof is only worth full marks when the reasoning is spelt out. The ratio calculations sit under Criterion A (Knowing and understanding), and scale-drawing or map problems reach into Criterion D (Applying maths in context).
Check yourself
1. Which congruence condition uses two sides and the angle between them? +
Two sides with the included angle is SAS (side, angle, side). The angle must be the one enclosed by the two named sides.
2. Two triangles are similar. A side of 5 cm on the small one matches 15 cm on the large one. Another small side is 4 cm; find its match. +
Scale factor \(= \dfrac{15}{5} = 3\). Multiply the other side: \(4 \times 3 = \mathbf{12}\) cm.
3. What kind of triangles does the RHS condition apply to, and what must match? +
RHS applies only to right-angled triangles. What must match is the right angle, the hypotenuse, and one other side. It is the special case that lets two sides plus a right angle prove congruence.
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