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!