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?

Saketh Sep 8, 2019

#1**+3 **

**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 \(\mathbf{97^\circ}\)

heureka Sep 8, 2019