The Largest Angle

Denote the three angles in the triangle by $A$ , $B$ , and $C$ . Then the three angles must add up to $180^{\circ}$ : $A+ B+ C = 180$ . Furthermore, the statement given in the problem can be translated to the equation $A = 2(B + C) - 24$ . We can solve the equation $A+ B+ C = 180$ for $C$ , getting $C = 180 - A - B$ , and substitute this in the other equation to get $A = 2(B + 180 - A - B) - 24$ . This equation simplifies to $A = 360 - 2A - 24$ , and then to $3A = 336$ , so we get $A = 112$ . Since this angle is greater than $90^{\circ}$ , it must be the largest angle. Therefore, the measure of the largest angle is 112 degrees.

$\begin{cases} \alpha + \beta + \gamma = 180 \\ \alpha = 2(\beta + \gamma)-24 \end{cases}$

Substitution of $\alpha$ in equation 1:

$2(\beta + \gamma)-24 + \beta + \gamma = 180$

$\implies 3(\beta + \gamma) = 204 \implies \beta + \gamma = \frac{204}{3} = 68$

Knowing that $\beta + \gamma = 68$ , it's obvious that $\alpha$ is the largest angle, and its value is $180-68 = \boxed{112^\circ}$