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.
Age
14 to 16
Challenge level
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
$$ \eqalign{
c^{\circ}&= 180^{\circ} - (a^{\circ}+b^{\circ})\cr &= 180^{\circ}-56^{\circ}\cr &=124^{\circ}}$$