Incentre Angle
Weekly Problem 1 - 2011
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 $\angle LOM$?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
Let
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
$$ \eqalign{
c^{\circ}&= 180^{\circ} - (a^{\circ}+b^{\circ})\cr &= 180^{\circ}-56^{\circ}\cr &=124^{\circ}}$$