Question

Given line AD is perpendicular bisector of line BC
Prove: Triangle ABD= ACD.

A. SSS
B. SAS
D. ASA
D. SSA

Answer 1

To show that triangles \( ABD \) and \( ACD \) are congruent, given that line \( AD \) is the perpendicular bisector of line \( BC \), we can use the **SAS** (Side-Angle-Side) congruence criterion.

### Proof:
1. **AD is the perpendicular bisector of BC**:
- This means that \( D \) is the midpoint of \( BC \). Thus, we have:
\[
BD = DC
\]
- Let’s denote this common length as \( x \):
\[
BD = DC = x
\]

2. **Angle formed by the perpendicular bisector**:
- Since \( AD \) is perpendicular to \( BC \), we have:
\[
\angle ADB = \angle ADC = 90^\circ
\]

3. **Common side**:
- Both triangles \( \triangle ABD \) and \( \triangle ACD \) share the side \( AD \):
\[
AD = AD
\]

### Conclusion using SAS:
- We now have:
- \( BD = DC \) (from the midpoint)
- \( \angle ADB = \angle ADC = 90^\circ \)
- \( AD = AD \) (common side)

Thus, by the SAS criterion, we conclude:
\[
\triangle ABD \cong \triangle ACD
\]

Therefore, the answer is **B. SAS**.