Question

$O$ is an interior point of $△ABC$; Let's prove that $\angle BOC\ge \angle BAC$

Answer 1

Let's draw a line AD through O.

$\angle BOD$ is the interior angle of $△AOB$

$\because$ Exterior angle=Sum of opposite interior angles.

$\angle BOD=\angle OAB+\angle OBA$

$\therefore \angle BOD\ge \angle OAB$-----------(1)

$\angle DOC$ is exterior angle of $△AOC$.

$\angle DOC=\angle OAC+\angle OCA$

$\therefore \angle DOC=\angle OAC$--------------(2)

Combining Equation(1) & (2), we get

$\angle BOD+\angle DOC\ge \angle OAB+\angle OAC$

$\therefore \angle BOC\ge \angle BAC$