Biggest of Smallest

Let $\beta$ and $\gamma$ be the other two angles of the triangle such that $\alpha \leq \beta \leq \gamma$ . Since the angles of a triangle sum to $180^\circ$ , thus

$\begin{aligned} 180^\circ &= \alpha + \beta + \gamma \\ &\geq \alpha + \alpha + \alpha \\ &\geq 3\alpha \\ \alpha &\leq 60^\circ \\ \end{aligned}$

Therefore, the maximum possible value of $\alpha$ is $60^\circ$ and we see this when the triangle is equilateral.

Let and be the other two angles of the triangle such that . Since the angles of a triangle sum to , thus Therefore, the maximum possible value of is and we see this when the triangle is equilateral.