Isometry Preserves Congruence

Theorem

Let $\Gamma = \R^n$ denote the real Euclidean space of $n$ dimensions, wher $n = 2$ or $n = 3$.

Let $\phi: \Gamma \to \Gamma$ be an isometry on $\Gamma$.

Let $\FF$ be a geometric figure in $\Gamma$.

The image of $\FF$ under $\phi$ is congruent to $\FF$, either directly or oppositely.

That is, $\phi$ preserves congruence.

Proof

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Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): isometry (isometric map)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): isometry (isometric map)