10-5 angles related to a circle and 10 6 more angle arc...
TRANSCRIPT
105 Angles Related to a Circle and 10 6 More angle arc theorems (1).notebook
1
April 28, 2017
Apr 212:59 PM
StarterFind the value of x.
1)
x
5 mm 2) x
20
15
3) Draw two circles that are externally tangent.4) Draw two circles that are internally tangent.
105 Angles Related to a Circle and 10 6 More angle arc theorems (1).notebook
2
April 28, 2017
Apr 279:37 AM
105 Angles Related to a Circle
Central Anglem <AOB = m Arc AB
A
O
B
Example:Given m<AOB = 72o, Find m Arc AB
105 Angles Related to a Circle and 10 6 More angle arc theorems (1).notebook
3
April 28, 2017
Apr 279:44 AM
Inscribed Anglem <DEF = 1/2 (m Arc DF)
D
E
F
G
H
I
Tangentchord Anglem <GHI = 1/2 (m Arc GH)
Example:Given m Arc DF = 84o, Find m <DEF
Example:Given m Arc GH = 160o, Find m <GHI
105 Angles Related to a Circle and 10 6 More angle arc theorems (1).notebook
4
April 28, 2017
Apr 279:50 AM
C
D
E
BA
Chordchord Anglem <DEC = 1/2 (m Arc AB + m Arc CD)m <AEB = 1/2 (m Arc AB + m Arc CD)
m <DEA = 1/2 (m Arc DA + m Arc CB)m <CEB = 1/2 (m Arc DA + m Arc CB)
Example 1:Given m Arc BC = 112o
m Arc AD = 186o Find m <DEA
Example 2:Given m <DEC = 50o
m Arc AB = 72o Find m Arc DC
105 Angles Related to a Circle and 10 6 More angle arc theorems (1).notebook
5
April 28, 2017
Apr 2812:28 PM
105 Angles Related to a Circle and 10 6 More angle arc theorems (1).notebook
6
April 28, 2017
Apr 279:56 AM
D
AP
B
C
Secantsecant Anglem <P = 1/2 (m Arc DC m Arc AB)
A
BC
D
Secanttangent Anglem <C = 1/2(m Arc AD m Arc BD)
Example:Given m Arc DC = 110o, m Arc AB = 32oFind m <P
Example:Given m Arc BD = 76o, m <C = 48oFind m Arc AD
105 Angles Related to a Circle and 10 6 More angle arc theorems (1).notebook
7
April 28, 2017
Apr 2711:12 AM
Tangenttangent Anglem <S = 1/2 (m Arc RQT m Arc RT)
.Q
R
T
S
Confused? See the summary on the next slide!(and p. 472 of your textbook!)
Example:Given m Arc RT = 104o, Find m <S
105 Angles Related to a Circle and 10 6 More angle arc theorems (1).notebook
8
April 28, 2017
Apr 2711:16 AM
SummaryIf the vertex is...
• AT THE CENTER the angle is equal to the intercepted arc
• ON THE CIRCLE the angle is half the intercepted arc• INSIDE THE CIRCLE (not at center) the angle is half the sum of the intercepted arcs
• OUTSIDE THE CIRCLE the angle is half the difference of the intercepted arcs
105 Angles Related to a Circle and 10 6 More angle arc theorems (1).notebook
9
April 28, 2017
Apr 3011:01 PM
Classwork• 10.5 Angles Related to a Circle Worksheet
105 Angles Related to a Circle and 10 6 More angle arc theorems (1).notebook
10
April 28, 2017
Apr 213:11 PM
10.6 More AngleArc Theorems
Theorem: If two inscribed or tangentchord angles intercept the same arc, then they are congruent.
A
B
X
YGiven: X and Y are inscribed
angles intercepting arc AB.
Conclusion:
Theorem: If two inscribed or tangentchord angles intercept congruent arcs, then they are congruent.
If ED is the tangent at D and
then we may conclude that
AP
BC
DE
105 Angles Related to a Circle and 10 6 More angle arc theorems (1).notebook
11
April 28, 2017
Apr 213:19 PM
Theorem: An angle inscribed in a semicircle is a right angle.
How can that be proven?
Theorem: The sum of the measures of a tangenttangent angle and its minor arc is 180.
How can that be proven?
The
A
BC
S
O
T P
O
105 Angles Related to a Circle and 10 6 More angle arc theorems (1).notebook
12
April 28, 2017
May 123:05 PM
Read the Sample Problems
on pages 480 481
105 Angles Related to a Circle and 10 6 More angle arc theorems (1).notebook
13
April 28, 2017
Apr 2711:32 AM
Homework• 10.5 Angles Related to a Circle Worksheet• p. 481 #3 5, 7, 10, 14
If you didn't finish in class
105 Angles Related to a Circle and 10 6 More angle arc theorems (1).notebook
14
April 28, 2017
Apr 2711:32 AM
Exit Slip
Name the point of concurrency shown.(incenter, circumcenter, orthocenter or centroid)
1. 2. 3.
Find the missing measures. SHOW WORK!
1. 2.
105 Angles Related to a Circle and 10 6 More angle arc theorems (1).notebook
15
April 28, 2017
Apr 213:06 PM
EXIT SLIP
105 Angles Related to a Circle and 10 6 More angle arc theorems (1).notebook
16
April 28, 2017
May 12:23 PM
105 Angles Related to a Circle and 10 6 More angle arc theorems (1).notebook
17
April 28, 2017
May 28:13 AM