The document defines and provides examples of central angles, inscribed angles, and the angles in a circle theorem. It states that a central angle is equal to twice the measure of the inscribed angle subtended by the same arc. Examples apply this theorem to find the measures of unspecified angles.

1. 1

2. 2

3. Minor Arc the smaller arc
.
C
A .
.
O
Major Arc .
B
the larger arc
3

4. Central Angle:
.
C
an angle that has it's vertex
at the centre, two radii form A .
the arms of the angle.
.
O
Central angle AOC is
subtended by the chord AC
and by the arc AC. .
B
4

5. Inscribed angle:
.
C
an angle that has it's vertex on
the circle, and two chords
form the arms.
A .
.
O
Inscribed angle ABC is
subtended by the chord AC, and
by the arc AC. .
B
5

6. 6

7. 7

8. Angles in a Circle Theorem
­ The angle inscribed in a semicircle is a right angle
­ The measure of the central angle is equal to twice
the measure of the inscribed angle subtended by
the same arc.
­ Inscribed angles subtended by the same arc, or
by equal arcs, are congruent.
8

9. Find the measure of angles
a a, and angle b.
b
.
O
50
o
Because the central angle is twice the inscribed if
they are subtended by the same arc
9

10. Find the measures of angles
o x, y, and z.
40
y
o
55
x
z
o
25
10

11. Exercise 31
Questions 1 ­ 7
11