Congruent & Similar 2D Shapes

The difference between congruent shapes (identical) and similar shapes (same shape, different size), how to spot each, and how to use proportion to find missing sides — with a clear diagram, worked examples and a quiz.

Two shapes can be related in two different ways: they can be exact copies of each other, or one can be a scaled photocopy of the other. Mathematics gives these two ideas precise names — congruent and similar.

1The difference in one picture

Congruent shapes are identical in every way. Similar shapes have the same angles but their sides are enlarged or reduced by the same factor.

2Congruent shapes

Two shapes are congruent if they have exactly the same shape and the same size.
Symbol: ≡

You can pick one up and place it exactly on top of the other — even if it has to be turned over or rotated first. All matching sides are equal and all matching angles are equal.

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Congruent shapes are still congruent after a translation, reflection or rotation — moving or flipping a shape never changes its size.

3Similar shapes

Two shapes are similar if they have the same shape but not necessarily the same size.
Symbol: |||

All matching angles are equal.
All matching sides are in the same ratio (the scale factor).

Every enlargement or reduction produces a similar shape. Note that all congruent shapes are also similar — with a scale factor of 1.

4Finding a missing side using proportion

Worked ExampleTwo similar triangles: the small one has sides 3 cm and 4 cm; the large one's matching side is 9 cm. Find its other side

1
Find the scale factor: 9 ÷ 3 = 3.
Divide a large side by its matching small side.
2
Multiply the other side by the same factor: 4 × 3 = 12.
Every side is enlarged by the same factor.
3
The missing side is 12 cm.
Check: 3:9 = 4:12 ✓

Worked ExampleAre these similar? Triangle A: 2 cm, 3 cm. Triangle B: 6 cm, 9 cm

1
Compare the ratios: 6 ÷ 2 = 3 and 9 ÷ 3 = 3.
Divide each pair of matching sides.
2
Both ratios are the same (3).
The sides are in proportion.
3
Yes — they are similar, with a scale factor of 3.

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Careful: shapes are only similar if every pair of matching sides has the same ratio. If one pair gives 2 and another gives 3, they are not similar.

Practice exercises

Work each one out, then click to reveal the answer.

📄 Worksheet ✅ Memo 📝 Exam paper 🗝️ Marking guide

1
What does it mean if two shapes are congruent?
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Same shape and same size — identical.
2
What does it mean if two shapes are similar?
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Same shape, different size: equal angles and sides in proportion.
3
Are all congruent shapes also similar?
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Yes — with a scale factor of 1.
4
Are all similar shapes congruent?
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No — only when the scale factor is 1.
5
Two similar triangles: sides 5 cm and 8 cm; the matching side to 5 is 15 cm. Find the other.
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Scale factor 3 → 8 × 3 = 24 cm
6
A rectangle 4 cm × 6 cm is enlarged by scale factor 2. What are the new sides?
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8 cm × 12 cm
7
Triangle A: 3, 5. Triangle B: 9, 15. Similar?
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9÷3 = 3 and 15÷5 = 3 → Yes, scale factor 3.
8
Triangle A: 2, 4. Triangle B: 6, 10. Similar?
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6÷2 = 3 but 10÷4 = 2,5 → No, ratios differ.