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 $a$, $b$, and $c$ using what you know about inscribed angles and the sum of the angles of a triangle.

   Hint (a):

   $88º = 2a$

   More Help (a):

   

   Look above at the image of the circle.
   $72º + d = 180º$ because they are supplementary angles.

   Hint 2(a):

   Use the Triangle Angle Sum Theorem to solve for $b$.

   $c = (2)(b)$

2. Compare the result for $c$ with $88º$ and $72º$. Is there a relationship?

   Answer (b):

   Since c is equal to $56º$, the measure of the vertical angles ($72º$) is the average of the measure of the two arcs that the vertical angles intercept.

   $\frac{88 + 56}{2} = 72º$