Question

if two straight lines intersects to each other then the opposite vertex angle are equal

Answer 1

yes opposite vertex angel are equal

Answer 2

yes, the opposite vertex angles will be equal because the angles vertically opposite to each other are called vertically opposite angles which are equal to each other.
proof-
mark the angles as ∠1,∠2,∠3 and ∠4.
∠1 +∠2=180 (linear pair)
∠3+∠2= 180 (linear pair)
therefore, ∠1+∠2=∠3+∠2
∠1+∠2-∠2=∠3+∠2-∠2                subtracting ∠2 on both sides
∠1=∠3
hence proved