The angles of triangle $ABC$ satisfy $\angle A < \angle B < \angle C$. The difference between $\angle B$ and $\angle A$ is equal to the difference between $\angle C$ and $\angle B$. If $\angle A = 23^\circ$, then what is $\angle C$, in degrees?

Guest Mar 3, 2020

#2**+1 **

The angles of triangle \(ABC\) satisfy \(\angle A < \angle B < \angle C\).

The difference between \(\angle B\) and \(\angle A\) is equal to the difference between \(\angle C\) and \(\angle B\).

If \(\angle A = 23^\circ\), then what is \(\angle C\), in degrees?

\(\begin{array}{|rcll|} \hline \mathbf{\angle B - \angle A} &=& \mathbf{\angle C - \angle B} \quad &| \quad A = 23^\circ \\ \angle B - 23^\circ &=& \angle C - \angle B \\ 2\angle B - 23^\circ &=& \angle C \\ \mathbf{2\angle B} &=& \mathbf{\angle C + 23^\circ} \\\\ \angle A + \angle B + \angle C &=& 180^\circ \quad &| \quad A = 23^\circ \\ 23^\circ + \angle B + \angle C &=& 180^\circ \\ \angle B + \angle C &=& 157^\circ \quad &| \quad *2 \\ 2\angle B + 2\angle C &=& 314^\circ \quad &| \quad \mathbf{2\angle B=\angle C + 23^\circ} \\ \angle C + 23^\circ + 2\angle C &=& 314^\circ \\ 3\angle C + 23^\circ &=& 314^\circ \\ 3\angle C &=& 314^\circ - 23^\circ \\ 3\angle C &=& 291^\circ \quad &| \quad :3 \\ \mathbf{ \angle C } &=& \mathbf{97^\circ} \\\\ 2\angle B - 23^\circ &=& \angle C \\ 2\angle B &=& 120^\circ \quad &| \quad :2 \\ \mathbf{ \angle B } &=& \mathbf{60^\circ} \\ \hline \end{array}\)

\(\angle C ~\text{ is }~ \mathbf{97^\circ}\)

heureka Mar 4, 2020