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The question for first is

In fig, it is given that AD = BC. By which Euclid's axiom it can be proved that AC = BD?

In fig state which lines are parallel and why?



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It is axiom 3.  If equals are subtracted from equals then the results are also equal.

AB - AD = AB - BC                subtracting equals  AD & BC

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.

1 5 1
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