## PROBLEM 3

Let the rays $[\mathrm{OA},[\mathrm{OB},[\mathrm{OC}$ and $[\mathrm{OD}$ be such that the angles $\angle A O B$ and $\angle B O C$ are adjacent, and the angles $\angle B O C$ and $\angle C O D$ are also adjacent. Consider the ray [OE as the bisector of $\angle A O B$, the ray [OF as the bisector of $\angle C O D$, and the ray [OS as the extension of the ray [OA. Given that the angles $\angle A O C$ and $\angle B O D$ are supplementary:

a) Determine the measure of the angle $\angle E O F$.

b) Prove that the angles $\angle D O S$ and $\angle B O C$ are congruent.
## Solution:

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$1 p$
$m(\angle A O B)+2 m(\angle B O C)+m(\angle C O D)=180^{\circ}$
$.1 p$
$2 m(\angle B O E)+2 m(\angle B O C)+2 m(\angle C O F)=180^{\circ}$ ..... $.1 p$
$m(\angle E O F)=m(\angle B O E)+m(\angle B O C)+m(\angle C O F)=90^{\circ}$
$1 p$
$m(\angle A O S)=180^{\circ}$
$1 p$
$m(\angle A O B)+m(\angle B O C)+m(\angle C O D)+m(\angle D O S)=180^{\circ}$ ..... $1 p$
$m(\angle A O B)+2 m(\angle B O C)+m(\angle C O D)=180^{\circ}$
$m(\angle B O C)-m(\angle D O S)=0 \rightarrow m(\angle B O C)=m(\angle D O S)$ ..... $1 p$