Rationality or otherwise of one angle included by two sides each having distinct integer lengths in an acute-angled scalene triangle

Question

To rephrase this question completely. If a triangle ABC has three different acute angles, ie no right angle or greater, and has two integer-length sides, the sides opposite the vertices A and B, say, and if the angle opposite the third side is $z\pi$, with $z>1/3$ must $z$ be a rational number or must it be an irrational number, or could it be either?

Thanks for all replies to the original question but please ignore that question, now. (It was misleading and read: Can a triangle include an irrational angle - by which I mean an angle in radians which is the product of an irrational number and $\pi$? It seems to me that if one angle is $z\pi$, with $z$ an irrational number, then the sum of this angle and either of the other two must be an irrational multiple of $\pi$ as well and therefore that the sum of all three must be be an irrational multiple of $\pi$. But the sum of the three angles must be $\pi$. Is there a flaw in my argument?)

Answer 1

Let $\alpha$ and $\beta$ be any positive real numbers such that $0<\alpha+\beta<1$. Then there is a triangle with angles $$\alpha\pi,\,\beta\pi,\,(1-\alpha-\beta)\pi$$

Answer 2

Why would that be a problem? Here is a triangle (which happens to be an isosceles one) in which all angles, when measured in radians, are irrational multiples of $\pi$: