在笛卡尔坐标系中，什么是等距变换？

As you have seen before, an isometry is a congruence transformation, such as translation. That is, an isometry does not change the figure itself, it just moves it.

All angles are a part of some triangle, and the image of the angle is one of the angles of the image of the triangle. By definition, an isometry preserves sidelengths, so by SSS, it should preserve angles as well.

Note that the definition does not include "preserves sidelengths and angles" but simply says "preserves sidelengths". This implies that it preserves angles as well. If it had said "preserves angles" then it may not necessarily preserve sidelengths (because we can scale our diagram without changing the angles, but all the sidelengths are multiplied by a common factor).

Suppose that we have a rectangle $ABCD$ with length $l$ and width $w$, and we map it by an isometry to another figure $A'B'C'D'$. What can we say about $A'B'C'D'$?

First, since isometries preserve distances, note the following: $$A'B'=AB=CD=C'D'$$ $$B'C'=BC=AD=A'D'$$ Since the pairs of opposite sides are still congruent, we are clearly dealing with a parallelogram.

Now, notice that $AC=A'C'$, and thus by SSS Congruence, $\triangle ABC\cong\triangle A'B'C'$. Therefore, we may deduce that $\angle A'B'C'$ is a right angle, just like $\angle ABC$.

Applying a similar logic to the other angles, we see that all of the angles are right, and thus, $A'B'C'D'$ is a rectangle congruent to $ABCD$.

Since both $ABCD$ and $A'B'C'D'$ are both rectangles with the same length and width, $$[ABCD]=[A'B'C'D']=lw$$

Thus, isometries preserve areas for all rectangles. A similar idea may be extended to all polygons: cut the polygons into triangles and repeatedly use SSS congruence.

Now that we have the result for rectangles, we may use the methods of calculus to finish our proof.

For more complicated shapes, simply approximate the shape by really thin rectangles. Make the rectangles thinner and thinner, and as they go to zero width, this approximation will become exact.

Since every rectangle will have its area preserved by the isometry, the total area of the shape will also be preserved. In fact, not only are areas preserved, but also shapes, since any figure will be mapped to a congruent figure, regardless of the isometry.