Given : Segment AC is perpendicular to BD
Segment BD bisects segment AC
Prove: Triangle ABD is congruent to Triangle CBD

Triangle ABD and CBD are congruent by the SAS postulate

Step-by-step explanation:

Step 1 :            

Given ,

Segment AC is perpendicular to BD

Segment BD bisects segment AC

To prove the given two triangles are congruent

Step 2 :

Since AC is perpendicular to BD , We have  ∠BDA and ∠BDC to be equal to 90 degrees

Also we have AD = DC , because BD bisects AC

BD is a side common to both the triangles and hence is the same

So we have two sides and angle between them (included angle)  to be same for both the triangles.

Hence Triangle ABD and CBD are congruent.