It is axiom 3. If equals are subtracted from equals then the results are also equal.
AB = AB
AB - AD = AB - BC subtracting equals AD & BC
DB = AC
So BD = AC
See diagram enclosed. Your diagram is not clear. not properly visible
1,2,3 are straight lines. 4,5,6 are angles.
angle 6 = angle 5.
Since angle 5 = angle 4, lines 1 and 2 are parallel. As they make same interior angle with straight line3.
Euclid's postulate 5 - parallel postulate.