BD is a perpendicular bisector of AC. Prove: Δ ABC is isosceles.

To prove that triangle ΔABC is isosceles, we can use the properties of the perpendicular bisector BD of segment AC.

1. Perpendicular Bisector: By definition, BD is a perpendicular bisector of AC. This means two things:

– It intersects AC at its midpoint D, which means ( AD = DC ).

– It forms right angles with AC, making angles ABD and CBD right angles.

2. Triangles ABD and CBD:

– Since BD serves as a common side for both triangles ABD and CBD, we have ( BD cong BD ) (common side).

– We already established that ( AD = DC ) (since D is the midpoint).

– Finally, ( angle ABD cong angle CBD ) (both are right angles).

3. Applying the SAS Criterion: By the Side-Angle-Side (SAS) postulate, we have:

– ( AD cong DC )

– ( BD cong BD )

– ( angle ABD cong angle CBD )

Thus, by SAS, ( triangle ABD cong triangle CBD ).

4. Conclusion: From this congruence, we can conclude that ( AB cong BC ). Since two sides of triangle ΔABC are equal, it follows that ΔABC is isosceles.

This proof shows that the triangles are