## SUBJECT III

Consider the adjacent angles $\mathrm{AOB}$ and $\mathrm{BOC}$ such that the measure of angle $\mathrm{BOC}$ is 5 times greater than the measure of angle $\mathrm{AOB}$. The bisector $\mathrm{OM}$ of angle BOC is constructed, $\mathrm{ON}$ is the opposite ray of ray $\mathrm{OM}$, and $\mathrm{DO}$ is perpendicular to $\mathrm{MN}$, with D and A on opposite sides of line $\mathrm{MN}$. Given that the measure of angle $\mathrm{AON}$ is $30^{\circ}$ more than twice the measure of angle $\mathrm{DOC}$, find the measures of angles $\mathrm{AOB}, \mathrm{BOC}, \mathrm{COD}$, and $\mathrm{AON}$.
## SUBJECT III

![](https://cdn.mathpix.com/cropped/2024_06_07_79bee740d14079536f8bg-3.jpg?height=431&width=491&top_left_y=296&top_left_x=725)

From the hypothesis, we have $\mathrm{m}(\nless \mathrm{BOC})=5 \cdot \mathrm{m}(\nless \mathrm{AOB})$....................................................................... $1 \mathrm{p}$

$\mathrm{m}(\nless \mathrm{COD})=90^{\circ}-\mathrm{m}(\nless \mathrm{MOC})$ which means $\mathrm{m}(\nless \mathrm{COD})=90^{\circ}-2.5 \cdot \mathrm{m}(\nless \mathrm{AOB}) \ldots . . . . . . . . . . . . . . . . . . . . . . .1 \mathrm{p}$

![](https://cdn.mathpix.com/cropped/2024_06_07_79bee740d14079536f8bg-3.jpg?height=62&width=1577&top_left_y=937&top_left_x=231)

![](https://cdn.mathpix.com/cropped/2024_06_07_79bee740d14079536f8bg-3.jpg?height=63&width=1579&top_left_y=1022&top_left_x=227)

![](https://cdn.mathpix.com/cropped/2024_06_07_79bee740d14079536f8bg-3.jpg?height=63&width=1570&top_left_y=1108&top_left_x=240)

![](https://cdn.mathpix.com/cropped/2024_06_07_79bee740d14079536f8bg-3.jpg?height=60&width=1582&top_left_y=1192&top_left_x=228)

$\mathrm{m}(\nless \mathrm{BOC})=100^{\circ}, \mathrm{m}(\nless \mathrm{COD})=40^{\circ}$ and $\mathrm{m}(\nless \mathrm{AON})=110^{\circ}$.................................................... $1 \mathrm{p}$

TOTAL ......................................................................................................................... 7p