Question

In Δ ABC, if AB is the greatest side, then prove that $\angle C>{60}^{\circ }$.

Answer 1

It is given that, AB is the longest side, so AB > BC and AB > AC.
Since, the angle opposite to the longer side is greater, therefore
$$\begin{gathered} \angle C>\angle A.....\left(i\right) \\ \angle C>\angle B.....\left(ii\right) \end{gathered}$$
Adding (i) and (ii), we get
$2\angle C>\angle A+\angle B$
Adding $\angle C$ on both sides, we have
$$\begin{gathered} 2\angle C+\angle C>\angle A+\angle B+\angle C \\ \Rightarrow 3\angle C>{180}^{\circ }\left(\because \angle A+\angle B+\angle C\right) \\ \Rightarrow \angle C>{60}^{\circ } \end{gathered}$$
Hence, $\angle C>{60}^{\circ }$.