Find Angle Q D3C993

1. **State the problem:** We need to find the measure of angle $q$ in the given figure where $AD$ and $CF$ are straight lines. 2. **Identify given angles and lines:** - $\angle FOE = 49^\circ$ - $\angle EOD = 68^\circ$ - $AD$ is a straight line through $O$ (so $\angle AOD = 180^\circ$) - $CF$ is a straight line through $O$ - $OA$ and $OB$ are perpendicular (right angle $90^\circ$) 3. **Calculate $\angle AOE$:** Since $AD$ is a straight line, $\angle AOD = 180^\circ$. $\angle AOE = \angle AOD - \angle EOD = 180^\circ - 68^\circ = 112^\circ$ 4. **Calculate $\angle FOE$ and $\angle FOC$ relationship:** Since $CF$ is a straight line, $\angle FOE + \angle EOC = 180^\circ$. Given $\angle FOE = 49^\circ$, so $\angle EOC = 180^\circ - 49^\circ = 131^\circ$ 5. **Calculate $\angle BOC$:** $OB$ is horizontal to the right, and $OA$ is vertical up, so $\angle AOB = 90^\circ$. $\angle BOC = \angle AOC - \angle AOB$ But $\angle AOC = \angle AOE + \angle EOC = 112^\circ + 131^\circ = 243^\circ$. Since angles around point $O$ sum to $360^\circ$, the angle $\angle BOC$ is the remaining part: $\angle BOC = 360^\circ - 243^\circ = 117^\circ$ 6. **Find $\angle q$:** $\angle q$ is the angle between $OB$ and $OC$, which is $\angle BOC = 117^\circ$. **Final answer:** $$\boxed{117^\circ}$$