In part (c) of problem 11-110, you used the relationship between the segment lengths formed by intersecting chords to find a missing length. But how are the arc measures of two random intersecting chords related? Examine the diagram at right.
, , and using what you know about inscribed angles and the sum of the angles of a triangle.
Look above at the image of the circle.
because they are supplementary angles.
Use the Triangle Angle Sum Theorem to solve for
Compare the result for
with and . Is there a relationship?
Since c is equal to
, the vertical angles ( ) are the average of the two arcs they intercept.