CPM Homework Help : CCG Problem 11-111

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.

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.

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 $b$ .
$c = (2)(∠b)$

Compare the result for $c$ with $88º$ and $72º$ . Is there a relationship?
Answer (b):
Since c is equal to $56º$ , the vertical angles ( $72º$ ) are the average of the two arcs they intercept.
$\frac{88 + 56}{2} = 72º$