Geometry/Trig 2 Name: __________________________Unit 8 GSP Explorations & Notes Date: ___________________________(Sections 9.2-9.6)

Section 9.2 Corollary

Theorem 9-4

Sketch the diagram:

Fill in the Measurements:

AB

BC

mAGB

mBHC

Conclusion (Theorem 9-4): In the same circle or in congruent circles, congruent chords intercept ___________________ arcs.

Examples: Find all angle and arc measures.

A

B

C

D

Q

Q is the center of the circle.

mAB = 86

mDC = _______ mDQC = ______

Classify DQC by sides: _____________

mBC = 128 mBAC = ___________

A

B

CmCAB = 40

mACB = ________ mABC = _______

mAB = 140

mAC = __________ mCB = _________

Conclusion (Corollary): Segments that are tangent to a circle from a point are ___________________.

Sketch the diagram:

Fill in the Measurements:

Example 1:B

C

AB and C are points of tangency.

What type of triangle is

BAC?

_______________________

mBAC = 32

mABC = ________

mBCA = ________Example 2:

A

C

B

4x + 2

½x + 9 B and C are points of tangency.

x = __________

BA = _________

CA = _________

BA

BC

Geometry/Trig 2 Name: __________________________Unit 8 GSP Explorations & Notes Date: ___________________________(Sections 9.2-9.6) – page 2

Theorem 9-5

Theorem 9-6

Sketch the diagram:

Fill in the Measurements:

Conclusion (Theorem 9-5): A diameter that is perpendicular to a chord _________________ the chord and its intercepted arc.

Examples (Q is the center of each circle).

RM

S

T

Q

P

15

17

RT = _________ QM = ________

QS = _________ MS = ________

SP = __________

Q

B

C

A

D

mADB = 220

mAB = ________ mAC = _________

mCB = _______ mAQC = ________

mAQB = _______ mABQ = ______

Challenge: If QC = 10, find AB.

FAF

FB

mAGC

mBHC

To measure the distance between a point and a segment, you must measure the _______________________________ distance.

Sketch the diagram:

Fill in the Measurements:

AD

AE

FG

CB

mFHG

mCKB

Conclusion (Theorem 9-6): In the same circle or in congruent circles, ___________________ chords are equally distant from the center.

Example (Q is the center of the circle).

P

K

J

N M

Q

L

Given: QJ = QL = 3; KP = 8

JP = _______ NM = _______

LM = _______ LN = ________

QM = _______ QK = ________

(d) mQNL = __________

You will need to draw in QM, QK, and QN to complete this problem.

Geometry/Trig 2 Name: __________________________Unit 8 GSP Explorations & Notes Date: ___________________________(Sections 9.2-9.6) – page 3

Theorem 9-7

Inscribed Angle: _________________________________________________________ ____________________________________________________________________________________________________________________________________________

Sketch the Diagram

Fill in the Measurements:

Section 9.5 Corollary 1

Conclusion (Theorem 9-7): The measure of an inscribed angle

is equal to ____________________________________________ of its

intercepted arc.

Example:

F

H

J

GmGFJ = ________

mHJ = __________

mFG = __________

mFGH = _________

mFHG = _________

92°44°

109°

Sketch the diagram:

Fill in the Measurements:

mABD

mACD

mAED

Conclusion (Corollary 1): Inscribed Angles that intercept the

same arc are ___________________________.

Example:

A

B

E

C

D

mAE = 102

mABE = __________

mACE = __________

mADE = __________

mBD = 129

mBAD = __________

mABC

mADC

mADCABCm

Geometry/Trig 2 Name: __________________________Unit 8 GSP Explorations & Notes Date: ___________________________(Sections 9.2-9.6) – page 4

Section 9.5 Corollary 2

Sketch the Diagram(include measurement):

Conclusion (Corollary 2): An angle inscribed inside of a semicircle is ___________________________________.

Examples: (AB is a diameter of each circle). (Round all decimal answers to the nearest tenth.)

B

Q

D

A

C

x°

y°mBD = 80

mADB = _____ mACB = _____

w = _________ x = __________ y

= _________ z = __________

w° z°

B A

D AB = 26, AD = 24, DB = ________

mDBA = ______ mDAB = _____

Section 9.5 Corollary 3

Sketch the Diagram(include four angle measurements):

Conclusion (Corollary 3): If a quadrilateral is inscribed in a

circle, then its opposite angles are _____________________.

Example:

Find:

mJKL = __________

mKLM = __________

mMJK = ___________

mJK = _____________

mMLK = ____________

mLMJ = ____________

mLMK = ____________

K

M

L

J

Complete:

AB is a _______________.

ACB is a _______________.

mLMJ = 73

mMJK = 88

mMJ = 102

Geometry/Trig 2 Name: __________________________Unit 8 GSP Explorations & Notes Date: ___________________________(Sections 9.2-9.6) – page 5

Theorem 9-8

Theorem 9-10 RULE: Angle = ½(Bigger Arc – Smaller Arc)

Case 1 – Two Secants Case 2 – Two Tangents Case 3 – A Secant & A Tangent

m1 = _________________ m2 = ________________ m3 = ________________

1

2 3

Example 1: Example 2:

A

C

B

DmCAB = 20

mDB = 115

mCB = _________

mCD = _________

mCDB = ________

mBCD = ________

A

C

D

B

mBC = 116

mBDC = ________

mCAB = _______

B is a point of tangency. B and C are points of tangency.

mBGD

mDBC

Sketch the Diagram:

Fill in the Measurements:

Conclusion (Theorem 9-8): The measure of an angle formed

by a chord and a tangent is equal to

__________________________ _____________________________ of the

intercepted arc.Example:

A B C

D

B is a point of tangency.

F

mDBC = 78

mDB = ____________

mDFB = ___________

mABD = __________

mBGDDBCm

Geometry/Trig 2 Name: __________________________Unit 8 GSP Explorations & Notes Date: ___________________________(Sections 9.2-9.6) – Answers to the Example Problems

Section 9.2 Corollary Theorem 9-4

Theorem 9-5 Theorem 9-6

Theorem 9-7 Section 9.5 Corollary 1

Section 9.5 Corollary 2 Section 9.5 Corollary 3

Example 1:mABC = 74

mBCA = 74

Example 2: x = 2

BA = 10

CA = 10

mDC = 86 mDQC = 86

Classify DQC by sides:

Isosceles

mBAC = 232

mACB = 70 mABC = 70

mAC = 140 mCB = 80

Example 1:

Example 2:

RT = 30 QM = 8

QS = 17 MS = 9 SP = 34

mAB = 140 mAC = 70

mCB = 70 mAQC = 70

mAQB = 140 mABQ = 20

Challenge: AB = 18.8

Example 1:

Example 2:

JP = 4 NM = 8

LM = 4 LN = 4

QM = 5 QK = 5

(d) mQNL = 36.9

mGFJ = 46

mHJ = 88

mFG = 71

mFGH = 251

mFHG = 289

mABE = 51

mACE = 51

mADE = 51

mBAD = 64.5

mADB = 90

mACB = 90

w = 40 x = 40

y = 50 z = 50AB = 26, AD = 24, DB = 10

mDBA = 67.4 mDAB = 22.6

Example 1:

Example 2:

mJKL = 107

mKLM = 92

mMJK = 184

mJK = 82

mMLK = 176

mLMJ = 214

mLMK = 278

Theorem 9-8mDB = 156

mDFB = 204

mABD = 102

Theorem 9-10

mCB = 75

mCD = 170

mCDB = 285

mBCD = 245

mBDC = 244

mCAB = 64