WarmUp

1
Warm-Up
A circle and an angle are drawn in the same
plane. Find all possible ways in which the
circle and angle intersect at two points.
2
9.4 Angles Formed by Secants and Tangents
Topics Theorems about measures of arcs
intercepted by angles
3
Classification of Angles with Circles
Case 1 Vertex is on the circle
Two secants
Secant and tangent
4
Classification of Angles with Circles
Case 2 Vertex is inside the circle
Two secants
5
Classification of Angles with Circles
Case 3 Vertex is outside the circle
Two secants
Secant and tangent
Two tangents
6
Vertex on Circle Secant and Tangent
Secant and tangent
7
Vertex on Circle Secant and Tangent
A
Part 1 The secant-tangent angle is a right
angle.
P
V
C
8
Vertex on Circle Secant and Tangent
Part 2 The secant-tangent angle is acute.
A
P
3
1
Copy and complete the table.
2
V
C
9
Vertex on Circle Secant and Tangent
X
Part 3 The secant-tangent angle is obtuse.
P
1
3
A
Copy and complete the table.
2
V
C
10
Theorem
If a tangent and a secant (or a chord) intersect
on a circle at the point of tangency, then the
measure of the angle formed is _______ the
measure of its intercepted arc.
one-half
11
Vertex Inside Circle Two Secants
Find the relationship between and
the measures of
A
D
2
1
V
P
B
Copy and complete the table.
C
12
Theorem
The measure of an angle formed by two secants or
chords that intersect in the interior of a circle
is _______ the _______ of the measures of the
arcs intercepted by the angle and its vertical
angle.
sum
one-half
13
Vertex Outside Circle Two Secants
Find the relationship between and the
measures of
B
2
A
1
V
C
Copy and complete the table.
D
14
Theorem
The measure of an angle formed by two secants
that intersect in the exterior of a circle is
________ the _________ of the measures of the
intercepted arcs.
one-half
difference
15
Example 1
Find m?DAB, m?CBA, and m?DGC.
Because the vertex of ?DAB is ON the circle,
.
m?DAB
16
Example 1
Find m?DAB, m?CBA, and m?DGC.
The vertex of ?CBA is outside the circle, so
m?CBA
17
Example 1
Find m?DAB, m?CBA, and m?DGC.
The vertex of ?DGC is inside the circle, so
m?DGC
18
Practice Problem
Find m?WXY, m?1, and m?2.
19
Homework
p.593 10-26