In the diagram, $O$ is the centre of circle $KLNM$ and $KO$ and $OM$ are joined - NSC Mathematics - Question 8 - 2017 - Paper 2

The angle $O_{1}$ can be found using the fact that the angle subtended at the centre ($O$) is twice that subtended at the circumference (point $N$) on the same arc. Therefore: $O_{1} = 2 imes N_{1} = 2 imes 80^{ ext{o}} = 160^{ ext{o}}$ Thus, the measure of angle $O_{1}$ is $160^{ ext{o}}$.

To find the angle $ilde{M}_{1}$, we can add up all the angles around point $O$. Given the angles are $360^{ ext{o}}$, $55^{ ext{o}}$, and $100^{ ext{o}}$, we calculate: $ilde{M}_{1} = 360^{ ext{o}} - (100^{ ext{o}} + 55^{ ext{o}} + 160^{ ext{o}}) = 360^{ ext{o}} - 315^{ ext{o}} = 45^{ ext{o}}$ Hence, the measure of angle $ilde{M}_{1}$ is $45^{ ext{o}}$.