Use the definition of a perpendicular bisector and notice that BD is a common side for both triangles.
Statements
Reasons
1.
BD⊥AC and BD bisects AC
1.
Given
2.
AD≅CD
2.
Definition of segment bisector
3.
m∠BDA=90^(∘) = m∠BDC
3.
Definition of perpendicular segments
4.
∠BDA ≅ ∠BDC
4.
Definition of congruent angles
5.
BD≅BD
5.
Reflexive Property of Congruent Segments
6.
△ ABD ≅ △ CBD
6.
Side-Angle-Side (SAS) Congruence Postulate
Practice makes perfect
We are given that BD⊥ AC, which implies that m∠ BDA= 90^(∘) = m∠ BDC. Also, since BD bisects AC we get AD≅CD. Let's highlight these facts in the given diagram.
Notice that side BD is common for both △ ABD and △ CDB. By the Reflexive Property of Congruent Segments we have BD ≅ BD. Next, let's summarize all we know about the triangles above.
cc
AD≅CD & Side
∠BDA≅∠BDC & Included Angle
BD≅BD & Side
Therefore, by the Side-Angle-Side (SAS) Congruence Postulate we conclude △ ABD ≅ △ CBD. Finally, let's write the two-column proof table.
