证明所有的等距变换保持角不变

We know that isometries, by definiton, are transformations that preserve distances.

Congruent triangles provide a simple and quick proof that isometries preserve angles.

Suppose that we have an angle $\angle ABC$. We map it by an isometry to the angle $\angle A'B'C'$.

Since the transformation is an isometry: $$AB=A'B'$$ $$BC=B'C'$$ $$CA=C'A'$$ By SSS congruence, we know that $\triangle ABC\cong\triangle A'B'C'$!

Now, since corresponding parts of congruent triangles are congruent (commonly abbreviated CPCTC), we have the desired result: that $\angle ABC\cong\angle A'B'C'$.

Since the choice of $\angle ABC$ was arbitrary, we can carry out the same process for any angle.

Thus, we see that all angles are preserved under isometries, and the proof is complete.