https://file.aiquickdraw.com/mathbot/file/1726755056771qxwoc3ap.jpg

To complete the proof for the given geometry problem, let's analyze the information step-by-step.

### Given:
- **\(\overrightarrow{AB}\) bisects \(\angle CAD\)**.
- **Prove: \(m \angle 1 = m \angle 3\)**.

### Diagram Explanation:
- \(\overrightarrow{AB}\) bisects the larger angle \(\angle CAD\), dividing it into two equal angles: \(\angle CAB\) and \(\angle BAD\).
- The angle numbers are as follows:
  - \(\angle 1\) is \(\angle CAB\),
  - \(\angle 3\) is \(\angle BAD\),
  - \(\angle 2\) is \(\angle BAC\).

### Proof Steps:

1. **Given Information**:  
   \(\overrightarrow{AB}\) bisects \(\angle CAD\). By the definition of an angle bisector, this means that:
   \[
   m\angle CAB = m\angle BAD
   \]
   Therefore, \(m \angle 1 = m \angle 3\).

2. **Linear Pair Postulate**:  
   \(\angle 2\) and \(\angle 1\) form a linear pair. Hence:
   \[
   m\angle 1 + m\angle 2 = 180^\circ
   \]
   Similarly, \(\angle 2\) and \(\angle 3\) also form a linear pair. So:
   \[
   m\angle 2 + m\angle 3 = 180^\circ
   \]

3. **Substitution**:  
   From the first equation, since \(m\angle 1 = m\angle 3\), substitute this into the second equation:
   \[
   m\angle 1 + m\angle 2 = 180^\circ
   \]
   This proves that the angles are equal, completing the proof.

Thus, **\(m \angle 1 = m \angle 3\)** as required.

---

Would you like more details or have any other questions? Here are five related questions:

1. What is the definition of an angle bisector?
2. How does the linear pair postulate help in proving equal angles?
3. Can this method be used to prove other angle equalities in different situations?
4. What other geometric principles could we use if the bisector was not given?
5. How do supplementary angles relate to the proof?

### Tip:
Whenever you encounter an angle bisector in a geometry problem, always think of dividing the larger angle into two equal parts, and look for linear pairs or supplementary angles.

Complete the proof: Given that AB bisects ∠CAD, prove that m∠1 = m∠3.

Solve the geometry problem where AB bisects ∠CAD and prove that m∠1 = m∠3 using the Angle Bisector Theorem and Linear Pair Postulate. Learn about key geometry concepts related to bisectors and angle relationships. Suitable for students in Grades 8-10.

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