12.2 HW Solutions3. 144. 25. 76. 507. 88. 109. a) CEb) DEc) angle CEBd) angle DEA10. The Center of the circle11. 612. 5.413. 8.914. 12.515. 9.916. 20.817. 10818. 9019. 123.9 or 124
23. a) PLb) PMc) All radii of a circle are congruentd) Triangle LPNe) SASf) CPCTC24. a) All radii of a circle are congruentb) AB ≅ CDc) Givend) SSSe) angle AEB ≅ CEDf) 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?
Player B
Player A
Player C
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
mACBm2
1 B
C
A
60°? ° ?°45 °
Three Cases for 12-9
Q R F
1) The center is on a side of the angle
2) The center is inside of the angle
3) The center is outside of the angle
1. BC is a diameter2. mAC = mAOC3. mAC = mA + mB4. mA = mB5. mAC =2mB6. ½mAC = mB
GivenDefinition of AC Tri. Ext. Ang. Thm.Isosceles Tri. ThmSubstitutionDivision Prop. Of Equal.
Given:
Prove:
Circle O, with inscribed B and diameter BC
mB = ½mAC
B
O
C
A
Find a and b
P
Q
R
S
T60˚
b˚
a˚
30˚
= 120°
= 75°
Corollaries: Inscribed Angle Theorem
1) Two inscribed angles that intercept the same arc are congruent
70˚
x
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
70˚
1
40˚
38˚
2
70˚
= 90°
= 38°
• Theorem 12-10– The measure of an angle formed by a tangent and
a chord is half the measure of the intercepted arc.
mBDCCm2
1
C
O
B
D
C
O
B
D
GEOGEBRA
Find x and y
K
L
QJ
35˚
y˚
x˚90°
= 55°
110°
70°
= 35°
Homework
• Section: 12-3• Pages: 681-683• Questions: 4-22, 36, 38
1. Circle O, inscribed ∠ABC2. Draw diameter BP3. mABP = ½AP4. mCBP = ½PC5. mABC = mABP + mABP 6. mABC = ½AP + ½PC7. mABC = ½(AP+PC)8. mABC = ½AC
GivenDef. of DiameterThm. 12-9Thm. 12-9Ang. Add. Post.SubstitutionDistributing (factoring)Arc Addition Post.
Given:
Prove: mABC = ½mAC
B
O
C
A
PCircle O, with inscribed ∠ABC
Post-Homework Review
• CHAPTER REVIEW• Page: 707• Questions: 6-14
• Section: 12-3• Pages: 681-683• Questions: 25
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