BD is a perpendicular bisector of AC. Prove: Δ ABC is isosceles.
ABD ≅ CBD
AB ≅ BC
AD ≅ DC
BD ≅ BD
BD is a perpendicular bisector of AC. Prove: Δ ABC is isosceles.
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BD is a perpendicular bisector of AC. Prove: Δ ABC is isosceles.
ABD ≅ CBD
AB ≅ BC
AD ≅ DC
BD ≅ BD
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