Denis said:
What's a "right n-angle"? Even Google is unaware of the term :shock:
It is necessary to use your imagination and develop a PLAN.
First, every n-angle is an n-gon. (It is probably one of the axioms that Hilbert introduced into geometry to make it rigorous because I do not remember it as a theorem in Euclid.)
Second, at least one of the angles in the desired n-gon must be a right angle.
Third, according to the question, the desired n-gon must be a REGULAR n-gon, or else "side" is ambiguous.
Fourth, it must have at least 6 sides, but it could be a regular right heptagon or regular right octagon etc . I must admit that MY imagination fails me at this step because 6 * 90 = 360 takes me into an arithmetic that I cannot quite fathom. (See, however, Gasking's essay on The Nature of Mathmatical Truth.)
Fifth, find the smallest integer k such that k > 5 and the regular right k-gon (pronounced klingon I believe) has the required property for its minimal and maximal diagonals.
Sixth, show (perhaps by mathmatical induction or reductio ad absurdum) that there is no integer m such that m > k and the regular right m-gon has the required property for its diagonals. (If k is not unique, the problem as stated is indeterminate.)
THERE. That is a plan for solving the given problem though it seems a bit difficult for a high-school student to carry out. Perhaps, therefore, the problem was not described quite accurately.