Euclidean plane/Improper isometry/Axis reflection/Fact

Let

$\varphi \colon \mathbb {R} ^{2}\longrightarrow \mathbb {R} ^{2}$

be an improper linear isometry.

Then ${}\varphi$ is an axis reflection,

and its describing matrix, with respect to the standard basis, has the form

${\begin{pmatrix}\cos \alpha &\sin \alpha \\\sin \alpha &-\cos \alpha \end{pmatrix}}$

with a uniquely determined angle ${}\alpha \in [0,2\pi [$.

Proof, Write another proof