Congruence and Similarity

Two big ideas

Geometry often asks whether two shapes are "the same". There are two precise versions of this:

• Congruent shapes are identical — same shape and same size. One is an exact copy of the other, even if it has been rotated, flipped or moved.
• Similar shapes are the same shape but a different size — one is a scaled-up or scaled-down copy of the other.

A quick way to remember: congruent = copy; similar = scaled.

Congruence in detail

If two shapes are congruent, then every matching side is equal and every matching angle is equal. You could cut one out and place it perfectly on top of the other (perhaps after turning or flipping it).

Proving triangles congruent

You do not need to check all six measurements. For triangles, any one of these four conditions is enough:

Condition	Meaning
SSS	All three sides equal
SAS	Two sides and the angle between them equal
ASA	Two angles and the side between them equal
RHS	Right angle, hypotenuse and one other side equal

Watch out: AAA is not a congruence condition. Three equal angles only prove the triangles are similar, because size is still free to change.

Described diagram: picture two triangles. The first has sides 5 cm, 6 cm, 7 cm. The second, drawn flipped over, also has sides 5 cm, 6 cm, 7 cm. By SSS, they are congruent — identical triangles, just mirrored.

Worked example: a congruence argument

Triangle ABC has AB = 8 cm, angle A = 40°, AC = 6 cm. Triangle PQR has PQ = 8 cm, angle P = 40°, PR = 6 cm. Are they congruent?

• Two sides (8 cm and 6 cm) and the angle between them (40°) are equal.
• That is the SAS condition.
• Therefore the triangles are congruent.

State the matching parts and name the condition — that is a complete proof.

Similarity in detail

Two shapes are similar when:

1. Matching angles are equal, and
2. Matching sides are in the same ratio — that ratio is the scale factor.

The scale factor tells you how many times bigger (or smaller) one shape is.

scale factor = new length ÷ original length

Worked example: finding a missing side

Two triangles are similar. The small triangle has sides 3 cm and 4 cm. The large triangle's side matching the 3 cm one is 9 cm. Find the side matching the 4 cm one.

1. Scale factor = 9 ÷ 3 = 3.
2. Matching side = 4 × 3 = 12 cm.

Every length in the large triangle is three times the matching length in the small one, while the angles stay exactly the same.

A useful warning about area

If lengths scale by a factor of k, then areas scale by k². So if a shape is enlarged by scale factor 3, its sides are 3 times longer but its area is 3² = 9 times larger. This catches many people out.

Activity: shadow scaling

Stand a metre ruler upright in sunlight and measure its shadow. At the same time, measure the shadow of a tree or tall pole. The ruler and the tall object form similar right-angled triangles with the sun's rays. Use the scale factor between the shadows to estimate the height of the tall object — real surveyors use exactly this trick.

Where this connects

Congruence relies on shapes that map onto each other through rotation and translation and reflection. Similarity, with its scale factors, leads naturally into enlargement and the side ratios used in introduction to trigonometry.