warm-up
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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. 9.4 – Angles Formed by Secants and Tangents. Topics: Theorems about measures of arcs intercepted by angles. Classification of Angles with Circles. - PowerPoint PPT PresentationTRANSCRIPT
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.
9.4 – Angles Formed by Secants and Tangents
Topics:
Theorems about measures of arcs intercepted by angles
Classification of Angles with Circles
Case 1: Vertex is on the circle
Two secants Secant and tangent
Classification of Angles with Circles
Case 2: Vertex is inside the circle
Two secants
Classification of Angles with Circles
Case 3: Vertex is outside the circle
Two secants Secant and tangent
Two tangents
Vertex on Circle – Secant and Tangent
Secant and tangent
Vertex on Circle – Secant and Tangent
A
C
m AVC
Part 1 – The secant-tangent angle is a right angle.
90 mAV 180
V
P
Vertex on Circle – Secant and Tangent
V
P
C
Part 2 – The secant-tangent angle is acute.
A1
2
3
Copy and complete the table.
mAV m 1 m 2 m PVC m AVC
120 120 30 60
100
80
x
Vertex on Circle – Secant and Tangent
V
P
C
Part 3 – The secant-tangent angle is obtuse.
A1
2
3Copy and complete the table.
mAXV m 1 m 2 m PVC m AVC
200 160 10 100
220
240
x
X
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
Vertex Inside Circle – Two Secants
VP
C
Find the relationship between and the measures of
A12
Copy and complete the table.
m AVC 1 and 2.
D
B
mAC mBD m 1 m 2 m AVC
160 40 80 100
180
2001x
m DVB20 100
70
602x
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.
one-half sum
Vertex Outside Circle – Two Secants
V C
Find the relationship between and the measures of A
1
2
Copy and complete the table.
mBD mAC m 1 m 2 m AVC
200 40 100 20
250
1001x
D
m 1 2 and AVC.
2x
80
60
50
B
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
Example 1
Find mDAB, mCBA, andmDGC.
Given: AB is tangent to circle Pat A, mAC = 60, mCD = 70,and mDE = 80
mDAB.
Because the vertex of DAB is ON the circle,
1
mAD2
1
60 702
65
Example 1
Find mDAB, mCBA, andmDGC.
Given: AB is tangent to circle Pat A, mAC = 60, mCD = 70,and mDE = 80
The vertex of CBA is outside the circle, so
mCBA 1
mEA mAC2
1
150 602
45
Example 1
Find mDAB, mCBA, andmDGC.
Given: AB is tangent to circle Pat A, mAC = 60, mCD = 70,and mDE = 80
The vertex of DGC is inside the circle, so
mDGC 1
mEA mCD2
1
150 702
110
Practice Problem
Find mWXY, m1, andm2.
Given: XY is tangent to circle Tat X, mUV = 90, mVX = 130,and mUW = 20 V
X
T
W Z
Y
1
2
Homework
p.593 #10-26