Download Inscribed Angles

12.2 HW Solutions
3.
4.
5.
6.
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2
7
50
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10
a) CE
b) DE
c) angle CEB
d) angle DEA
The Center of the circle
6
5.4
8.9
12.5
9.9
20.8
108
90
123.9 or 124
23. a) PL
b) PM
c) All radii of a circle are congruent
d) Triangle LPN
e) SAS
f) CPCTC
24. a) All radii of a circle are congruent
b) AB ≅ CD
c) Given
d) SSS
e) angle AEB ≅ CED
f) Thm 12.1
Inscribed Angles
Section 12.3
Three high-school soccer players practice kicking goals from the
points shown in the diagram. All three points are along an arc
of a circle. Player A says she is in the best position because the
angle of her kicks towards the goal is wider than the angle of
the other players’ kicks. Do you agree?
Vocabulary
• Inscribed Angle
– An angle whose vertex is on the circle and whose
sides are chords of the circle
• Intercepted Arc
– The portion of the circle intercepted by an
inscribed angle
• Theorem 12-9: Inscribed Angle Theorem
– The measure of an inscribed angle is half the
measure of its intercepted arc
A
1
mB  mAC
2
B
?°°
45
60°
?°
C
Three Cases for 12-9
Q
1) The center is on
a side of the angle
R
2) The center is
inside of the angle
F
3) The center is
outside of the angle
Given:
Circle O, with inscribed B and diameter BC
Prove:
mB = ½mAC
C
A
O
B
1. BC is a diameter
2. mAC = mAOC
3. mAC = mA + mB
4. mA = mB
5. mAC =2mB
6. ½mAC = mB
Given
Definition of AC
Tri. Ext. Ang. Thm.
Isosceles Tri. Thm
Substitution
Division Prop. Of Equal.
Find a and b
P
a˚ = 120°
T
60˚
30˚
Q
S
b˚ = 75°
R
Corollaries: Inscribed Angle Theorem
1) Two inscribed angles that intercept the same arc
are congruent
x
70˚
Corollaries: Inscribed Angle Theorem
2) An angle inscribed in a semicircle is a right angle
( A semicircle is 180, ½ of that is 90)
x° = 90°
Corollaries: Inscribed Angle Theorem
3) The opposite angles of a quadrilateral inscribed in a
circle are supplementary
(opposite angles intercept the entire circle, ½ of 360 is 180)
70˚
x
Corollaries: Inscribed Angle Theorem
1) Two inscribed angles that intercept the same arc
are congruent
2) An angle inscribed in a semicircle is a right angle
( A semicircle is 180, ½ of that is 90)
3) The opposite angles of a quadrilateral inscribed in a
circle are supplementary
(opposite angles intercept the entire circle, ½ of 360 is 180)
Find the measure of angles 1 and 2
40˚
2 = 38°
70˚
38˚
70˚
1 = 90°
• Theorem 12-10
– The measure of an angle formed by a tangent and
a chord is half the measure of the intercepted arc.
1
mC  mBDC
2
B
B
D
D
O
O
C
GEOGEBRA
C
Find x and y
110°
Q
J
35˚
90°
x˚ = 35°
70°
y˚ = 55°
L
K
Homework
• Section: 12-3
• Pages: 681-683
• Questions: 4-22, 36, 38
P
Given: Circle O, with inscribed ∠ABC
Prove: mABC = ½mAC
A
O
B
1. Circle O, inscribed ∠ABC
2. Draw diameter BP
3. mABP = ½AP
4. mCBP = ½PC
5. mABC = mABP + mABP
6. mABC = ½AP + ½PC
7. mABC = ½(AP+PC)
8. mABC = ½AC
Given
Def. of Diameter
Thm. 12-9
Thm. 12-9
Ang. Add. Post.
Substitution
Distributing (factoring)
Arc Addition Post.
C
Post-Homework Review
• CHAPTER REVIEW
• Page: 707
• Questions: 6-14
• Section: 12-3
• Pages: 681-683
• Questions: 25