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Home > CCG > Chapter 11 > Lesson 11.2.3 > Problem 11-111


2 intersecting chords, divides circle into 4 arcs, labeled as follows: left, 88 degrees, right, c. Angle on left side of intersection, labeled, 72 degrees. Dashed segment, connecting bottom ends of chords, creates triangle, with left bottom angle labeled, b, and right bottom angle labeled, a.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.  

  1. Solve for , and  using what you know about inscribed angles and the sum of the angles of a triangle.

    Added to diagram, angle below intersection, labeled, d.

    Look above at the image of the circle.
    because they are supplementary angles.

    Use the Triangle Angle Sum Theorem to solve for .

  1. Compare the result for with  and . Is there a relationship?

    Since c is equal to , the measure of the vertical angles () is the average of the measure of the two arcs that the vertical angles intercept.