## Subject III.

Lines AB and CD intersect at O. Knowing that rays [OM, [OT, [OR, are the bisectors of angles $\angle \mathrm{BOD}, \angle \mathrm{DOM}$, and respectively, $\angle \mathrm{COB}$, and $\mathrm{m}(\angle \mathrm{AOM})=130^{\circ}$, calculate:
a) $\mathrm{m}(\angle \mathrm{ROM})$;
b) $\mathrm{m}(\angle \mathrm{AOD})$;

c) the measure of the angle between the bisectors of angles $\angle \mathrm{AOC}$ and $\angle \mathrm{BOR}$.
## SUBJECT III

Lines AB and CD intersect at O. Knowing that rays [OM, [OT, [OR, are the bisectors of angles $<\mathrm{BOD},<\mathrm{DOM}$, and respectively, $<\mathrm{COB}$, and $\mathrm{m}(<\mathrm{AOM})=130^{\circ}$, calculate:
a) $\mathrm{m}(<\mathrm{ROM})$;
b) $\mathrm{m}(<\mathrm{AOD})$;

c) the measure of the angle between the bisectors of angles $<$ AOC and $<$ BOR.

Diagram

$$
\begin{aligned}
& \mathrm{m}(<\mathrm{ROM})=90^{\circ} \\
& \mathrm{m}(<\mathrm{BOM})=50^{\circ}= \mathrm{m}(<\mathrm{DOM}) ; \mathrm{m}(<\mathrm{AOD})=80^{\circ} \\
& \text { angle sought }= 180^{\circ}-1 / 2^{*} \mathrm{~m}(<\mathrm{AOC})-1 / 2^{*} \mathrm{~m}(<\mathrm{ROB})= \\
&=180^{\circ}-1 / 2^{*} \mathrm{~m}(<\mathrm{DOB})-1 / 2 * 1 / 2^{*} \mathrm{~m}(<\mathrm{COB})= \\
&=180^{\circ}-1 / 2^{*} 100^{\circ}-1 / 4^{*} 80^{\circ}=110^{\circ}
\end{aligned}
$$