Question

What is the reason that AX is perpendicular to XC

Answer 1

The statement doesn't tell us why AX is perpendicular to XC, but we actually don't need to know that.  All we need to show is that if AX is perpendicular to XC, then $\angle AXC$ is right.

Luckily, this is true because of the definition of perpendicular lines.  Two perpendicular lines form right angles, so if two perpendicular segments intersect, they form a right angle at their intersection.  So, if AX is perpendicular to XC, angle AXC is right.