What is the range of angle in front of longest triangle edge?

Question

What is the minimum and the maximum values of the angle $\gamma$ in front of the longest triangle edge?

Answer 1

For the minimum :

Let $\alpha\le \beta\le \gamma$ be the inner angles of a triangle.

Suppose that $\gamma\lt 60^\circ$. Then, $$180^\circ=\alpha+\beta+\gamma\lt 60^\circ+60^\circ+60^\circ=180^\circ,$$ which is a contradiction.

Hence, we have $\gamma\ge 60^\circ$. The equality is attained when a triangle is an equilateral triangle.