How to prove the congruency of these triangles

Question

In the below sketch, we have angle $\beta$ given, and a right angle $\angle CEB$ and a right angle $\angle CED$. The problem asked to show that $\triangle CEB$ is congruent to $\triangle CED$.

They use ASA for that, by saying that because $\beta$ is equal to $\angle DCE$ without further justification, but is that so obvious from the given information? It's obvious of course why ASA works after we can find $\angle DCE = \beta$, but it's not so clear to me how to justify this claim.

Answer 1

$\beta$ is equal to $\angle DCE$ ... but it's not so clear to me how to justify this claim

Angles $\beta$ and $\angle DCE$ are called vertical angles, and they are equal because they have the same supplement $\angle ACE\,$. The proof goes all the way back to Euclid's Proposition I.15: "If two straight lines cut one another, then they make the vertical angles equal to one another".

Answer 2

There are two angles marked as $ \beta$

Note that the angle DCE is also $\beta$ therefore ASA works fine.