## SUBJECT III

Let angles $\Varangle A O B$ and $\varangle B O C$ be such that $m(\varangle B O C)$ is four times smaller than $\mathrm{m}\left(\nless \mathrm{AOB}\right)$. If $\mathrm{m}(\nless \mathrm{AOC})=120^{\circ}$, and (OD is the opposite ray of (OB, determine $\mathrm{m}(\varangle \mathrm{AOD})$. ( Analyze all possible situations).
## Solution:

If we denote $m(\nless B O C)=x$, then $m(\varangle A O B)=4 x$

$1 p$

![](https://cdn.mathpix.com/cropped/2024_06_07_f344a73543ba19f21944g-3.jpg?height=68&width=1610&top_left_y=736&top_left_x=301)

![](https://cdn.mathpix.com/cropped/2024_06_07_f344a73543ba19f21944g-3.jpg?height=62&width=1437&top_left_y=820&top_left_x=469)

Case II: Angles $\Varangle$ AOB and $\Varangle$ BOC are adjacent, then we have the following situations:

![](https://cdn.mathpix.com/cropped/2024_06_07_f344a73543ba19f21944g-3.jpg?height=65&width=1442&top_left_y=973&top_left_x=466)

![](https://cdn.mathpix.com/cropped/2024_06_07_f344a73543ba19f21944g-3.jpg?height=60&width=1388&top_left_y=1046&top_left_x=520)

ii) $m(\varangle A O C)+m(\varangle B O C)+m(\varangle B O A)=360^{\circ} \Rightarrow 5 x=240^{\circ} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots$

Finalization: $x=48^{\circ} \Rightarrow m(\nless A O B)=192^{\circ}>180^{\circ}$ - does not fit ..................1p