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Geometry
10.4 Other Angle Relationships in Circles
Thursday, March 23, 2:46
Geometry 10.4 Other Angle Relationships in Circles 2
Goal
Use angles formed by tangents, secants, and chords to solve problems.
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Geometry 10.4 Other Angle Relationships in Circles 3
Review
115 4 9
21
2 15 2 4 92
30 4 9
39 4
399.75
4
x
x
x
x
x
Note: in solving an equation with fractions, one of the first things to do is always “clear the fractions”.
Thursday, March 23, 2:46
Geometry 10.4 Other Angle Relationships in Circles 4
You do it. Solve: 120 4 10
4x
120 4 10
41
4 20 4 4 104
80 4 10
90 4
22.5
x
x
x
x
x
Thursday, March 23, 2:46
Geometry 10.4 Other Angle Relationships in Circles 5
Review
The measure of an inscribed angle is equal to one-half the measure of the intercepted arc.
8040
What if one side of the angle is tangent to the circle?
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Geometry 10.4 Other Angle Relationships in Circles 6
Theorem 10.2: Tangent-Chord
A
BC
12
If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one-half the measure of the intercepted arc.
1 122 and1 2m mA m m CAB B
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Geometry 10.4 Other Angle Relationships in Circles 7
Simplified Formula
ab
12
12
12
1
2
m a
m b
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Geometry 10.4 Other Angle Relationships in Circles 8
Example 1
Find the and .mAB mBCA
1280
160
mAB
mAB
A
BC
80
360 160
200
mBCA
160200
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Geometry 10.4 Other Angle Relationships in Circles 9
Example 2. Solve for x.
A
BC
4x
(10x – 60)
124 (10 60)
8 10 60
2 60
30
x x
x x
x
x
Thursday, March 23, 2:46
Geometry 10.4 Other Angle Relationships in Circles 10
If two lines intersect a circle, where can the lines intersect each other?
On the circle.
Inside the circle.
Outside the circle.
We already know how to do this.
Thursday, March 23, 2:46
Geometry 10.4 Other Angle Relationships in Circles 11
Theorem 10.13 (Inside the circle)
A
B
C
D
1
If two chords intersect in a circle, then the measure of the angle is one-half the sum of the intercepted arcs.
121m mAB mCD
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Geometry 10.4 Other Angle Relationships in Circles 12
Simplified Formula
1a
b
121m a b
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Geometry 10.4 Other Angle Relationships in Circles 13
Example 3 Find m1.
A
B
C
D
130
80 1
2
12
1 30 80
(110)
55
m
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Geometry 10.4 Other Angle Relationships in Circles 14
Example 4 Solve for x.
A
B
C
D
6020
x
1260 20
120 20
100
x
x
x
100
Check:
100 + 20 = 120
120 ÷ 2 = 60
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Geometry 10.4 Other Angle Relationships in Circles 15
Your turn. Solve for x & y.
A
B
C
D
x75 85M
y
K
P
O
20
32
132 20
264 20
44
y
y
y
175 85
280
x
x
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Geometry 10.4 Other Angle Relationships in Circles 16
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Geometry 10.4 Other Angle Relationships in Circles 17
Secant-Secant
C
A
BD
121m mAB mCD
1
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Geometry 10.4 Other Angle Relationships in Circles 18
Simplified Formula
1 b a
121m a b
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Geometry 10.4 Other Angle Relationships in Circles 19
Secant-Tangent
C
A
B
121m mAB mBC
1
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Geometry 10.4 Other Angle Relationships in Circles 20
Simplified Formula
a
b1
121m a b
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Geometry 10.4 Other Angle Relationships in Circles 21
Tangent-Tangent
A
B
121m mACB mAB
1
C
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Geometry 10.4 Other Angle Relationships in Circles 22
Simplified Formula
1 ab
121m a b
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Geometry 10.4 Other Angle Relationships in Circles 23
Example 5 Find m1.
1 8010
1 12 21 80 10 (70) 35m
35
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Geometry 10.4 Other Angle Relationships in Circles 24
Example 6 Find m1.
1
12070
1 12 21 120 70 (50) 25m
25
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Geometry 10.4 Other Angle Relationships in Circles 25
Example 7 Find m1.
1
210
150
1 12 21 210 150 (60) 30m
30?
360 – 210 = 150
k
m
Rays k and m are tangent to the circle.
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Geometry 10.4 Other Angle Relationships in Circles 26
How to remember this: If the angle vertex is on the circle, its
measure is one-half the intercepted arc.
If an angle vertex is inside the circle, its measure if one-half the sum of the intercepted arcs.
If an angle vertex is outside the circle, its measure is one-half the difference of the intercepted arcs.
![Index [link.springer.com]978-1-4757-2133-1/1.pdf · Chvatal, Linear programming, simplex, 458 Ci, 19 circles angle between geometry[find_angle], 580 Appolonius circles geometry[Appolonius],](https://img.pdfslide.net/doc/110x75/5ad9de017f8b9afc0f8bb749/index-link-978-1-4757-2133-11pdfchvatal-linear-programming-simplex-458.jpg)
