Parallel Lines and Circle Arcs

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

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.

the parallel lines can be
on the same side

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?

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:

• Draw a diagonal line (a transversal) from an endpoint of a 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 create equal arcs