We are given that:

• $m \angle AOC = 135^\circ$
• $m \angle BOC = 64^\circ$

We are tasked with finding $m \angle AOB$.

Approach:

1. Notice that the points $A, B, C, D$ are connected through a common point $O$, and the angles are measured from this point.
2. $\angle AOC$ and $\angle AOB$ share the common side $OA$, while $\angle BOC$ is between $OB$ and $OC$.
3. The full circle around point $O$ must sum up to $360^\circ$, and we can use this to solve for $\angle AOB$.

Step-by-step solution:

1. Angle sum around point $O$: The angles $\angle AOC + \angle BOC + \angle AOB$ should sum up to $360^\circ$, as they form part of the angles around point $O$.

   $m \angle AOC + m \angle BOC + m \angle AOB = 360^\circ$

2. Substitute known values: $135^\circ + 64^\circ + m \angle AOB = 360^\circ$

3. Solve for $m \angle AOB$: $m \angle AOB = 360^\circ - (135^\circ + 64^\circ)$ $m \angle AOB = 360^\circ - 199^\circ$ $m \angle AOB = 161^\circ$

Final Answer:

$m \angle AOB = 161^\circ$

Let me know if you need further details or clarifications!

Related Questions:

1. How can we use complementary angles to solve problems like these?
2. What is the total sum of angles in a circle, and how does it apply here?
3. How would this problem change if more angles around $O$ were given?
4. How can we determine the angle between lines intersecting at a point in general?
5. What is the relationship between adjacent angles in a problem like this?

Tip: When dealing with angles around a point, always remember that the sum of angles around a point is $360^\circ$.