Parallel Lines and Circle Arcs

When two parallel lines cut through a circle, the arcs they trap between them are equal:

Circle intersected by two horizontal parallel lines, highlighting the equal arcs trapped between them

The arcs between the parallel lines are equal in measure.

It doesn't matter if the lines are horizontal, vertical, or at an angle. If they are parallel, the arcs on the "sides" will have the same measure.

Circle with two parallel lines on the upper half, showing that the intercepted arcs are still equal

the parallel lines can be
on the same side

Circle with a parallel tangent line and secant line, showing equal intercepted arcs

and can be tangent

This also works if a line just touches the edge of the circle (a tangent). The two arcs from the touch-point down to the other line will still be equal.

Why is this so?

Circle enclosing an isosceles trapezoid formed by connecting the endpoints of parallel chords

Think about connecting the crossing points ... we get a symmetrical isosceles trapezoid ... the arcs it cuts out must be identical.

Or use Alternate Interior Angles:

Circle with parallel chords and a diagonal transversal line showing equal alternate interior angles

Draw a transversal line from an endpoint of one chord to the opposite endpoint of the other chord
Because the lines are parallel, the angles created at the corners are equal
In a circle, equal inscribed angles always intercept equal arcs, so the two arcs must be equal