Geometry/Trig 2Name: __________________________ Unit 8 GSP Explorations & NotesDate: ___________________________ (Sections ) 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 = _______ m  DQC = ______ Classify  DQC by sides: _____________ mBC = 128mBAC = ___________ A B C m  CAB = 40 m  ACB = ________ m  ABC = _______ 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 A B and C are points of tangency. What type of triangle is  BAC? _______________________ m  BAC = 32 m  ABC = ________ m  BCA = ________ Example 2: A C B 4x + 2 ½x + 9 B and C are points of tangency. x = __________ BA = _________ CA = _________ BA BC

Geometry/Trig 2Name: __________________________ Unit 8 GSP Explorations & NotesDate: ___________________________ (Sections ) – 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). R M S T Q P RT = _________ QM = ________ QS = _________ MS = ________ SP = __________ Q B C A D mADB = 220 mAB = ________ mAC = _________ mCB = _______ m  AQC = ________ m  AQB = _______ m  ABQ = ______ Challenge: If QC = 10, find AB. F AF 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). Given: QJ = QL = 3; KP = 8 JP = _______ NM = _______ LM = _______ LN = ________ QM = _______ QK = ________ (d) m  QNL = __________ You will need to draw in QM, QK, and QN to complete this problem.

Geometry/Trig 2Name: __________________________ Unit 8 GSP Explorations & NotesDate: ___________________________ (Sections ) – 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 G m  GFJ = ________ mHJ = __________ mFG = __________ mFGH = _________ mFHG = _________ 92° 44° 109° Sketch the diagram: Fill in the Measurements: m  ABD m  ACD mAED Conclusion (Corollary 1): Inscribed Angles that intercept the same arc are ___________________________. Example: mAE = 102 m  ABE = __________ m  ACE = __________ m  ADE = __________ mBD = 129 m  BAD = __________ m  ABC mADC

Geometry/Trig 2Name: __________________________ Unit 8 GSP Explorations & NotesDate: ___________________________ (Sections ) – 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.) x° y° mBD = 80 m  ADB = _____ m  ACB = _____ w = _________ x = __________ y = _________ z = __________ w°z° B A D AB = 26, AD = 24, DB = ________ m  DBA = ______ m  DAB = _____ 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: m  JKL = __________ m  KLM = __________ mMJK = ___________ mJK = _____________ mMLK = ____________ mLMJ = ____________ mLMK = ____________ Complete: AB is a _______________. ACB is a _______________. m  LMJ = 73 m  MJK = 88 mMJ = 102

Geometry/Trig 2Name: __________________________ Unit 8 GSP Explorations & NotesDate: ___________________________ (Sections ) – page 5 Theorem 9-8 Theorem 9-10RULE: Angle = ½(Bigger Arc – Smaller Arc) Case 1 – Two SecantsCase 2 – Two TangentsCase 3 – A Secant & A Tangent m  1 = _________________m  2 = ________________m  3 = ________________ 1 23 Example 1:Example 2: A C B D m  CAB = 20 mDB = 115 mCB = _________ mCD = _________ mCDB = ________ mBCD = ________ A C D B mBC = 116 mBDC = ________ m  CAB = _______ B is a point of tangency. B and C are points of tangency. mBGD m  DBC 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 BC D B is a point of tangency. F m  DBC = 78 mDB = ____________ mDFB = ___________ m  ABD = __________

Geometry/Trig 2Name: __________________________ Unit 8 GSP Explorations & NotesDate: ___________________________ (Sections ) – Answers to the Example Problems Section 9.2 CorollaryTheorem 9-4 Theorem 9-5Theorem 9-6 Theorem 9-7Section 9.5 Corollary 1 Section 9.5 Corollary 2Section 9.5 Corollary 3 Example 1: m  ABC = 74 m  BCA = 74 Example 2: x = 2 BA = 10 CA = 10 mDC = 86 m  DQC = 86 Classify  DQC by sides: Isosceles mBAC = 232 m  ACB = 70 m  ABC = 70 mAC = 140 mCB = 80 Example 1: Example 2: RT = 30 QM = 8 QS = 17 MS = 9 SP = 34 mAB = 140mAC = 70 mCB = 70m  AQC = 70 m  AQB = 140 m  ABQ = 20 Challenge: AB = 18.8 Example 1: Example 2: JP = 4NM = 8 LM = 4LN = 4 QM = 5QK = 5 (d) m  QNL = 36.9 m  GFJ = 46 mHJ = 88 mFG = 71 mFGH = 251 mFHG = 289 m  ABE = 51 m  ACE = 51 m  ADE = 51 m  BAD = 64.5 m  ADB = 90 m  ACB = 90 w = 40x = 40 y = 50z = 50 AB = 26, AD = 24, DB = 10 m  DBA = 67.4 m  DAB = 22.6 Example 1: Example 2: m  JKL = 107 m  KLM = 92 mMJK = 184 mJK = 82 mMLK = 176 mLMJ = 214 mLMK = 278 Theorem 9-8 mDB = 156 mDFB = 204 m  ABD = 102 Theorem 9-10 mCB = 75 mCD = 170 mCDB = 285 mBCD = 245 mBDC = 244 m  CAB = 64