Idiot\'s Guides Basic Math and Pre-Algebra

202 Part 3: The Shape of the World

Suppose that in circle O, OA and OB are radii and AC and BC are chords. If AB 50 , find

m’AOB and m’ACB.

AB is the intercepted arc for both angles. m’AOB is a central angle, so its measure is the same

as the measure of the arc. m’AOB = 50r. m’ACB is an inscribed angle, so its measure is half the

measure of the arc m’ACB = 25r.

WORLDLY WISDOM

An angle inscribed in a semicircle is a right angle.

When two chords intersect within a circle, they form four angles, which are labeled with

numbers in the figure. Vertical angles are congruent, so ’ 1 # ’3 and ’ 2 # ’4. You might look

at the picture and think that arc AC is smaller than arc BD (and you’d be right), and so you might

wonder how the two angles could have the same measurement.

Draw chord AD to make a triangle. ’4 is an exterior angle of that triangle, and so it’s equal to

m’DAB + m’CDA. Those are inscribed angles, so m’4 = m’DAB + m’CDA =

1

2 AC +

1

2 DB.

A B

C

O

A

(^12)
(^43)
B
D
C