When the measures of the angles of a triangle are placed in order, the difference between the middle angle and smallest angle is equal to the difference between the middle angle and largest angle. If one of the angles of the triangle has measure 23 degrees, then what is the measure in degrees of the largest angle of the triangle?

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  1. clearly, 23 is not the largest. If it is the smallest, then

    23 + 23+d + (23+d)+d = 180
    3d + 69 = 180
    3d = 111
    d = 37

    and the angles are 23,60,97

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  2. Let the middle angle have measure $x$, and let the difference between this measure and each of the others be $y$. So, the angle measures are $x-y, x, x+y$. These must sum to $180^\circ$ because they are the measures of the angles of a triangle. So, we must have
    \[(x-y)+x + (x+y) = 180^\circ.\]Simplifying the left-hand side gives $3x = 180^\circ$, so $x = 60^\circ$ is the measure of the middle angle. Therefore, the $23^\circ$ is the smallest angle, and is $60^\circ - 23^\circ = 37^\circ$ less than the middle angle. So, the largest angle has measure $60^\circ + 37^\circ = \boxed{97^\circ}$.

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