I am not understanding this math about alternate angles

Question

This is the math. This is the given solution :-

This

and this

Now, my problem here is, $\angle ABC$ and $\angle DAC$ are supposed to be alternate angles here, and thus equal.

But as far as I know, angles are alternate only when a transversal intersects two parallel lines.

But in the given figure, there are no such transversals or parallel lines.

So how can we prove that, $\angle ABC = \angle DAC$?

Answer 1

I think what the solution meant to refer to was the alternate segments theorem:

The angle between a chord of a circle and a line tangent to the circle at one end of the chord is equal to the angle of the alternate segment.

The angle $\angle ABC$ is the angle in "the alternate segment" because it is the angle subtended by the chord $AC$ at $B,$ because $B$ is in one of the segments into which the chord $AC$ cuts the circle, and because the segment $B$ is in is the one that is not included in the angle $\angle DAC$ (making it the "alternate" segment).