+128 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, then what is the measure in degrees of the largest angle of the triangle?
+26367 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, then
what is the measure in degrees of the largest angle of the triangle?
\(\text{Let $\alpha+\beta+\gamma=180^\circ$ with $\alpha < \beta < \gamma$ } \)
\(\begin{array}{|rcll|} \hline \gamma-\beta &=& \beta-\alpha \\ \mathbf{\gamma} &=& \mathbf{2\beta-\alpha} \quad | \quad \alpha=180^\circ-\beta-\gamma \\ \gamma &=& 2\beta-(180^\circ-\beta-\gamma) \\ \gamma &=& 2\beta-180^\circ+\beta+\gamma \quad | \quad -\gamma \\ 0 &=& 2\beta-180^\circ+\beta \\ 180^\circ &=& 3\beta \\ \beta &=& \dfrac{180^\circ}{3} \\ \mathbf{\beta} &=& \mathbf{60^\circ} \\ \hline \end{array}\)
\(\begin{array}{|rcll|} \hline \gamma &=& 2\beta-\alpha \quad | \quad \alpha = 23^\circ,\ \beta=60^\circ \\ \gamma &=& 2\cdot 60^\circ-23^\circ \\ \gamma &=& 120^\circ-23^\circ \\ \mathbf{\gamma} &=& \mathbf{97^\circ} \\ \hline \end{array}\)
The measure in degrees of the largest angle of the triangle is \(\mathbf{97^\circ}\)
