Idiot\'s Guides Basic Math and Pre-Algebra

Chapter 15: Circles 203

But then you notice that ’2 is also an exterior angle of that triangle and equal to exactly the
same thing. The measure of an angle formed by two chords (and the measure of its vertical angle
partner) is one-half the sum of the two intercepted arcs. To find the measure of an angle formed
by two chords, average the arcs intercepted by the two vertical angles.

If two chords intersect in the circle as shown, and you want to find the value of x, you just need
to realize that the two vertical angles whose measure is x intercept arcs of 40r and 50r. Then
x^1
2

40 50 45.

A

x x

50 40

B

E

D

C

CHECK POINT

6. ’TOP is a central angle in circle O, and arc TP measures 48r. Find the measure
   of ’TOP.


7. Points M, A, and N lie on circle O. Arc MN measures 78r. Find the measure of
   ’MAN.


8. Diameter SI meets chord IT at point I on circle P. Arc TS measures 36r. Find
   the measure of ’SIT.


9. In circle P described in question 8, chords IT and ST meet each other at point
   T and meet the ends of diameter SI. Find the measure of ’ITS.


10. Chords LT and GP intersect at point A inside circle O. Arc LG measures 56r and
    arc PT measures 82r. Find the measure of ’GAL and the measure of ’PAL.