# Definition:Similar Figures

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## Definition

Two rectilineal figures are **similar** if and only if:

- They have corresponding angles, all of which are equal
- They have corresponding sides, all of which are proportional.

### Informal Definition

Two geometric figures are **similar** if they have the same *shape* but not necessarily the same size.

It is intuitively understood what it means for two figures to have the same shape.

### Algebraic Definition

Two geometric figures are **similar** if one can be transformed into the other by means of a similarity mapping.

### Euclid's Definition

In the words of Euclid:

**Similar rectilineal figures**are such as have their angles severally equal and the sides about the equal angles proportional.

(*The Elements*: Book $\text{VI}$: Definition $1$)

## Also see

- Results about
**similar figures**can be found**here**.

## Historical Note

The symbol introduced by Gottfried Wilhelm von Leibniz to denote geometric similarity was the tilde: $\sim$

This is still in use and can still be seen, but is not universal.

## Sources

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**similar** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**similar** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**similar**(of figures)