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In this problem, we are asked to find the measure of angle \( \angle C \) in triangle \( \triangle ABC \), given the following information:

- \( \overline{AD} \) bisects \( \angle CAB \).
- \( m\angle ADB = 96^\circ \).
- \( m\angle CAB = 64^\circ \).

### Step-by-step solution:

1. **Angle Bisector Property:**
   Since \( AD \) bisects \( \angle CAB \), it divides \( \angle CAB \) into two equal angles. Therefore, 
   \[
   m\angle BAD = m\angle CAD = \frac{m\angle CAB}{2} = \frac{64^\circ}{2} = 32^\circ.
   \]

2. **Sum of Angles in \( \triangle ADB \):**
   We know that the sum of interior angles in any triangle is \( 180^\circ \). In triangle \( \triangle ADB \), the angles are \( \angle BAD = 32^\circ \), \( \angle ADB = 96^\circ \), and \( \angle ABD \). The sum of these angles must be:
   \[
   m\angle BAD + m\angle ADB + m\angle ABD = 180^\circ.
   \]
   Substituting the known values:
   \[
   32^\circ + 96^\circ + m\angle ABD = 180^\circ,
   \]
   \[
   m\angle ABD = 180^\circ - 128^\circ = 52^\circ.
   \]

3. **Sum of Angles in \( \triangle ABC \):**
   Now, let's consider the entire triangle \( \triangle ABC \). The sum of the interior angles in this triangle is also \( 180^\circ \). We know \( m\angle CAB = 64^\circ \) and \( m\angle ABC = 52^\circ \). Let \( m\angle ACB = m\angle C \).

   The equation for the sum of the angles is:
   \[
   m\angle CAB + m\angle ABC + m\angle ACB = 180^\circ,
   \]
   \[
   64^\circ + 52^\circ + m\angle ACB = 180^\circ,
   \]
   \[
   m\angle ACB = 180^\circ - 116^\circ = 64^\circ.
   \]

### Final Answer:
The measure of \( \angle C \) (or \( m\angle ACB \)) is \( 64^\circ \).

### Explanation:
We used the fact that the sum of angles in a triangle is always \( 180^\circ \), along with the information about angle bisectors and the given angles to solve for \( \angle C \).

Would you like more details on any part of this solution?

Here are some related questions for deeper understanding:
1. What is the angle bisector theorem and how does it apply in this case?
2. Why do the interior angles of a triangle always sum to \( 180^\circ \)?
3. Can this approach be used for all triangles, or only specific types?
4. How would the problem change if \( \overline{AD} \) were not an angle bisector?
5. How can we verify that \( \triangle ABC \) is a scalene triangle based on the given angles?

**Tip:** Remember, angle bisectors create two equal angles by definition, and this is often key to solving geometric problems involving triangles!

In the diagram, AD bisects ∠CAB, m∠ADB = 96° and m∠CAB = 64°. Find m∠C and explain how you know.

Learn how to find the measure of angle C in triangle ABC using the angle bisector property and the sum of interior angles. The problem includes details like m∠ADB = 96° and m∠CAB = 64°. Suitable for Grades 9-10.

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