https://file.aiquickdraw.com/mathbot/file/1729523485817930r765r.jpg

The proof provided aims to demonstrate that an angle has been bisected in the given diagram. Let's go through the proof step-by-step and fill in the missing steps.

### Problem: Prove that the angle has been bisected.

### Given Information:
- A quadrilateral is drawn with two triangles \( \triangle ABD \) and \( \triangle BCD \).
- The task is to prove that the angle has been bisected.

### Steps filled in so far:

1. **\( \triangle BAD \cong \triangle BCD \)**  
   (Reason: Triangles are congruent by some criterion, likely Side-Angle-Side or another rule, depending on the diagram's given conditions.)

2. **\( \angle ABD \cong \angle CBD \)**  
   (Reason: Corresponding angles of congruent triangles are equal.)

3. **\( \angle B = \angle ABD + \angle CBD \)**  
   (Reason: Angle Addition Postulate, which states that the whole angle is the sum of its parts.)

4. **\( \angle B = \angle ABD + \angle ABD \)**  
   (Reason: Substitution Property, where we replace \( \angle CBD \) with \( \angle ABD \) because they are congruent.)

5. **\( \frac{1}{2} \angle B = \angle ABD \)**  
   (Reason: Multiplication Property of Equality, since both halves of angle \( B \) are congruent, we can divide by 2.)

This completes the steps given in the image. Essentially, we have proven that \( \angle ABD \) is half of \( \angle B \), meaning that angle \( B \) has been bisected by line \( BD \).

### Questions to expand understanding:

1. What criteria are used to prove that triangles \( \triangle BAD \) and \( \triangle BCD \) are congruent (SAS, SSS, ASA)?
2. Can you explain why corresponding angles of congruent triangles must be equal?
3. How does the Angle Addition Postulate apply to the angle bisector proof?
4. Could this proof be applied to other geometric shapes or only quadrilaterals?
5. What would change if line \( BD \) did not bisect \( \angle B \)?

### Tip:
When proving angles or segments are bisected, always look for congruent triangles and use properties like substitution and the angle addition postulate.

In this geometric proof, we demonstrate that an angle has been bisected using congruent triangles and the properties of angle addition and equality. The steps involve proving triangle congruence and applying the Angle Addition Postulate.

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