If you construct a pair of perpendicular lines, they intersect such that a pair of circles with a finite radius exist where each circle is tangent at two points to either line. Any circle tangent to both lines has a centre lying on the line bisecting one of the right angles.

As the circle radius increases (and the centre gets farther from the point of intersection of the pair of lines) the perpendicular lines are tangent to the circle 90 degrees apart, and as the radius approaches infinity, the points of tangency remain at zero degrees to the circle. But a circle with infinite radius has a straight line between any two points on it, so at infinity the two tangent lines are parallel and 180 degrees apart.