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Inscribed Angles

Learn to find and prove measures of inscribed angles in chapter 10-4, exploring various properties of circles, polygons, and parallel lines. Includes proofs and exercises for students.

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Inscribed Angles

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  1. Inscribed Angles Chapter 10-4

  2. Find measures of inscribed angles. • Find measures of angles of inscribed polygons. • intercepted Standard 7.0Students prove and use theorems involving the properties of parallel lines cut by a transversal, the properties of quadrilaterals, and the properties of circles. (Key) Standard 21.0Students prove and solve problems regarding relationships among chords, secants, tangents, inscribed angles, and inscribed and circumscribed polygons of circles. (Key) Lesson 4 MI/Vocab

  3. Inscribed Angles Def: An angle whose vertex is on a circle and whose sides contain chords • The arc that lies in the interior of an inscribed angle and has endpoints on the angle is called the Intercepted Arc of the angle

  4. Y X Inscribed Angles • The measure of an inscribed angle is one half of its intercepted arc. X = ½ Y Y = 2X

  5. a a 108 Measures of Inscribed Angles 20 40 20 + 40 + a + 108 + a = 360 2a + 168 = 360 2a = 192 a = 96 Lesson 4 Ex1

  6. 108 Measures of Inscribed Angles 20 40 m1 = m2 = m3 = m4 = m5 = 96 96 180 – (20 + 54) = 106 Lesson 4 Ex1

  7. A • B • C • D A. 30 B. 60 C. 15 D. 120 Lesson 4 CYP1

  8. A • B • C • D A. 110 B. 55 C. 125 D. 27.5 Lesson 4 CYP1

  9. A • B • C • D A. 30 B. 80 C. 40 D. 10 Lesson 4 CYP1

  10. A • B • C • D A. 110 B. 55 C. 125 D. 27.5 Lesson 4 CYP1

  11. A • B • C • D A. 110 B. 55 C. 125 D. 27.5 Lesson 4 CYP1

  12. Inscribed Angles • If two inscribed angles of a circle intercept the same arc (or  arcs), then the angles are . X Y mX = m Y

  13. Given: Proof with Inscribed Angles Prove:ΔPJKΔEHG

  14. Proof: Statements Reasons 1. 1. Given 2. If 2 chords are , corr. minor arcs are . 2. 3. Definition of intercepted arc 3. 4. 4. Inscribed angles of arcs are . 5. 5.Right angles are congruent. 6. ΔPJK ΔEHG 6.AAS Proof with Inscribed Angles Lesson 4 Ex2

  15. Given: Choose the best reason to complete the following proof. Prove:ΔCEMΔHJM Lesson 4 CYP2

  16. Proof: Statements Reasons 1. 2. 3. 4. 5. ΔCEMΔHJM 1. Given 2. ______ 3. Vertical angles are congruent. 4. Radii of a circle are congruent. 5.ASA • Alternate Interior Angle Theorem • Substitution C. Definition of  angles D. Inscribed angles of  arcs are . Lesson 4 CYP2

  17. Inscribed Arcs and Probability This Topic will not be on the chapter 10 test. But Geometric probability will appear on the CST. 9th and 10th grade teachers should cover this topic briefly. Lesson 4 Ex3

  18. The probability that is the same as the probability of L being contained in . Inscribed Arcs and Probability This Topic will not be on the chapter 10 test. But Geometric probability will appear on the CST. 9th and 10th grade teachers should cover this topic briefly. Lesson 4 Ex3

  19. A. B. C.D. • A • B • C • D This Topic will not be on the chapter 10 test. But Geometric probability will appear on the CST. 9th and 10th grade teachers should cover this topic briefly. Lesson 4 CYP3

  20. A C B Inscribed Angles • A 90o inscribed angle intercepts a 180o arc • C is a right   AB is a diameter of the circle R = 2.5 5 3 4

  21. Lesson 4 TH3

  22. Lesson 4 Ex4

  23. A • B • C • D A. 45 B. 90 C. 180 D. 80 Lesson 4 CYP4

  24. A • B • C • D A. 17 B. 76 C. 60 D. 42 Lesson 4 CYP4

  25. A • B • C • D A. 17 B. 76 C. 60 D. 42 Lesson 4 CYP4

  26. A • B • C • D A. 73 B. 30 C. 60 D. 48 Lesson 4 CYP4

  27. Inscribed Polygons • Reminder: If all the vertices of a polygon lie on the circle • The polygon is inscribed in the circle • The circle is circumscribed about the polygon • A quadrilateral can be inscribed in a circle  its opposite angles are supplementary • Supplementary—adds up to 180o

  28. Inscribed Polygons (3x + 2) + (2x –7) =180 5x – 5 = 180 5x = 185 x = 37 3x + 2 2x-7

  29. Angles of an Inscribed Quadrilateral Draw a sketch of this situation. Lesson 4 Ex5

  30. To find we need to know To find first find Angles of an Inscribed Quadrilateral Inscribed Angle Theorem Sum of arcs in circle=360 Subtract 174 from each side. Lesson 4 Ex5

  31. Angles of an Inscribed Quadrilateral Inscribed Angle Theorem Substitution Divide each side by 2. Since we now know three angles of a quadrilateral, we can easily find the fourth. mQ + mR + mS + mT = 360 360° in a quadrilateral 87 + 102 + 93 + mT = 360 Substitution mT = 78 Subtraction Answer:mS = 93; mT = 78 Lesson 4 Ex5

  32. A • B • C • D A. 99 B. 104 C. 81 D. 76 Lesson 4 CYP5

  33. A • B • C • D A. 99 B. 104 C. 81 D. 76 Lesson 4 CYP5

  34. Homework Chapter 10-4 • Pg 583 6 – 8, 15 – 21, 24 – 31 all

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