Let $\triangle A B C$ be a triangle such that the height relative to side $B C$ is not less than side $B C$ and the height relative to side $A B$ is not less than side $A B$. Determine the measures of the angles of this triangle.
Let $h_{a}$ and $h_{c}$ be the altitudes relative to the sides $BC=a$ and $AB=c$, respectively. By hypothesis, we have $h_{a} \geq a$ and $h_{c} \geq c$. Since $h_{a}$ and $h_{c}$ are the lengths of the altitudes, then $h_{a} \leq c$ and $h_{c} \leq a$.

One of the sides considered is greater than or equal to the other: say $a \geq c$. From the inequalities above, we have:

$$
h_{a} \geq a \geq c \geq h_{a} \quad \Rightarrow \quad a=c=h_{a}
$$

Thus, it follows that $AB$ is perpendicular to $BC$. Therefore, the triangle is a right isosceles triangle, and the angles measure $45^{\circ}, 45^{\circ}$, and $90^{\circ}$.