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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 \( 360^\circ \).

If m∠AOC = 135° and m∠BOC = 64° then what is m∠AOB?

This geometry problem involves calculating m∠AOB using the angle sum theorem. Given that m∠AOC = 135° and m∠BOC = 64°, the total angle around the point is 360°. This problem is suitable for students in Grades 7-9.

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