A circle is the set of all points that are a given distance from a fixed point.
The given distance is called the radius. The fixed point is called the center.
5
A circle is the set of all points that are a given distance from a fixed point.
The given distance is called the radius. The fixed point is called the center.
Here are two points. Which one is part of the circle?



A circle is the set of all points that are a given distance from a fixed point.
The given distance is called the radius. The fixed point is called the center.
Secant is a line that intersects
a circle in two points.
Chord is a segment that joins
two points on a circle.
Diameter is a chord that passes through the center.
Definition: A tangent is a line that meets a circle in exactly one point, called
the point of tangency.

Definition: A tangent is a line that meets a circle in exactly one point, called
the point of tangency.
Theorem: If a line is tangent to a circle, then the line is perpendicular to the
radius drawn to the point of tangency.
Proof: Suppose AP
is not perpendicular
to tangent line l.
P
Q
A
l
In the diagram, segments PB and PC are tangent to circle A.
Prove that PB and PC are congruent.
B
P
A
C
Theorem: The tangent segments drawn to a circle from an external
point are congruent.
An arc is part of a circle. A
F
D
B
E
C
An arc is the set of points consisting
of 2 points on a circle and all the
points on the circle between them.
G
H
Semicircle ABC
Minor Arc DE
FHG
Majo r Arc FGH
An arc is part of a circle. A
F
D
B
E
C
An arc is the set of points consisting
of 2 points on a circle and all the
points on the circle between them.
G
H
Semicircle ABC
Minor Arc DE
FHG
Majo r Arc FGH
Central angle of a circle is an angle whose vertex
is the center of the circle.
E
The measure of a minor arc is defined to be
the measure of its central angle. The measure
of a circle is 360
F
Theorem: In the same circle or in congruent circles.
(1) Congruent central angles have congruent arcs
(2) Congruent arcs have congruent central angles
(3) Congruent chords have congruent arcs
(4) Congruent arcs have congruent chords.
G
<EFG is a central angle.

Its measure is equal to the
measure of minor arc EG.
mEFG = mEG
Theorem: A diameter that is perpendicular to a chord, bisects the chord and its arcs.
3
Theorem: A _______
diameter that is perpendicular
to a chord, bisects the chord and its arcs.
___________________
1
2
Any “2x2 converse” is also true.
4
An inscribed angle is an angle whose vertex is on a circle and whose sides contain
chords of the circle.
Theorem: The measure of an inscribed angle is equal
to half the measure of its intercepted arc.
BAD = ½ (BD)
An inscribed angle is an angle whose vertex is on a circle and whose sides contain
chords of the circle.
Theorem: The measure of an inscribed angle is equal
to half the measure of its intercepted arc.
Theorem: If two inscribed angles intercept the same arc,
then they are congruent.
BAD = ½ (BD)
A
D
In the diagram, BAC  BDC because
they are both inscribed angles and they
intercept the same arc.
C
B
Theorem: An angle inscribed in a semicircle is a right angle.
Theorem: An angle formed by two chords intersecting
inside a circle is equal to half the sum of
the intercepted arcs.
ARE = ½ (AE + DC)
Theorem: An angle formed by two chords intersecting
inside a circle is equal to half the sum of
the intercepted arcs.
ARE = ½ (AE + DC)
Find the measure of
1. AED
65
2. CAE
60
3. BFA
55
B
20
C
60
50
F
D
A
160
E
Theorem: An angle formed by two secants, two tangents, or by a secant and a
tangent drawn from a point outside a circle is equal to half the
difference of the intercepted arcs.
APE = ½ (AE – CB)
Theorem: An angle formed by two secants, two tangents, or by a secant and a
tangent drawn from a point outside a circle is equal to half the
difference of the intercepted arcs.
APE = ½ (AE – CB)
APE = ½ (AE – AC)
APE = ½ (ANE – AE)
Find the values of x and y.
x=____ ; y=_____
x = 31
155
4x
x
y
y = 46.5
Theorem: The measure of an angle formed by a tangent and a chord of a
circle is half the measure of the arc between them.
ABC = ½ AB
B
C
A
Triangle ABC is an isosceles triangle with AB = AC.
Find the measures of x, y, and z.
A
x = 140
40
y = 70
x
z = 40
y
C
z
B
EF is tangent to circle P; AD is a diameter;
AB = 30, CD = 40, DE =50
1 = 40
Find the measure of each numbered angle.
2 = 15
3 = 25
4 = 40
E
F
6
5
9
5 = 65
50
2
130
10
4
P
1
3
A
30
8
H
7 = 45
40
8 = 30
C
9 = 25
10 = 65
7
B
D
6 = 100
110
In the diagram, radius AB is perpendicular to radius AC. Point E is
chosen randomly on minor arc BC, and CD is constructed perpendicular
to ray BE at point D.
Using Geometer’s Sketchpad, make a conjecture about the relationship
between the lengths of segments CD and DE and prove you are correct.
B
E
D
A
C