# Incentre Angle

Weekly Problem 1 - 2011

Use facts about the angle bisectors of this triangle to work out another internal angle.

Use facts about the angle bisectors of this triangle to work out another internal angle.

## Problem

The three angle bisectors of triangle $LMN$ meet at a point $O$ as shown.

$\angle LNM$ is $68^{\circ}$. What is the size of $\angle LOM$?

Image

If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.

## Student Solutions

Image

Let

$\angle OLM = \angle OLN =
a^{\circ},$

$\angle OML = \angle OMN = b^{\circ}$
and

$\angle LOM = c^{\circ}$

Angles in a triangle add up to $180^{\circ}$, so from $\triangle LMN$, $$2a^{\circ}+2b^{\circ}+68^{\circ} = 180^{\circ}$$ which gives $$ 2(a^{\circ}+b^{\circ})=112^{\circ}$$ In other words $$a^{\circ}+b^{\circ}=56^{\circ}$$

Also, from $\triangle LOM$, $$a^{\circ}+b^{\circ}+c^{\circ}=180^{\circ}$$ and so

$$ \eqalign{

c^{\circ}&= 180^{\circ} - (a^{\circ}+b^{\circ})\cr &= 180^{\circ}-56^{\circ}\cr &=124^{\circ}}$$