I received a similar question from the 2023 UKMT Intermediate Mathematical Challenge asking that of an equilateral triangle, quadrilateral and pentagon, which, if any, could be have all different angles.
I reckoned that it was only the pentagon and then proved it after the challenge on geogebra. How would I have proved that it was only the pentagon, during the challenge with only pen and paper?
Here's a physical model argument that's pretty convincing.
Build a plane equilateral pentagon $ABCDE$ with hinges at the vertices. Fix edge $AE$. You clearly have two degrees of freedom to rotate edges $AB$ and $DE$ independently about line $AE$. That will change the angle at $C$ continuously.
In all but a very few special configurations no two angles will be equal.
This argument can be made rigorous. That said, I would cheerfully accept it as written in an Intermediate Mathematical Challenge.