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Inscribed Angle Theorem: Find Angle Measures in Inscribed Triangles

Learn how to use inscribed angles to find angle measures in inscribed triangles. Includes proofs and real-world examples.

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Inscribed Angle Theorem: Find Angle Measures in Inscribed Triangles

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  1. Splash Screen

  2. Five-Minute Check (over Lesson 10–3) CCSS Then/Now New Vocabulary Theorem 10.6: Inscribed Angle Theorem Proof: Inscribed Angle Theorem (Case 1) Example 1: Use Inscribed Angles to Find Measures Theorem 10.7 Example 2: Use Inscribed Angles to Find Measures Example 3: Use Inscribed Angles in Proofs Theorem 10.8 Example 4: Find Angle Measures in Inscribed Triangles Theorem 10.9 Example 5: Real-World Example: Find Angle Measures Lesson Menu

  3. A. 60 B. 70 C. 80 D. 90 5-Minute Check 1

  4. A. 60 B. 70 C. 80 D. 90 5-Minute Check 1

  5. A. 40 B. 45 C. 50 D. 55 5-Minute Check 2

  6. A. 40 B. 45 C. 50 D. 55 5-Minute Check 2

  7. A. 40 B. 45 C. 50 D. 55 5-Minute Check 3

  8. A. 40 B. 45 C. 50 D. 55 5-Minute Check 3

  9. A. 40 B. 30 C. 25 D. 22.5 5-Minute Check 4

  10. A. 40 B. 30 C. 25 D. 22.5 5-Minute Check 4

  11. A. 24.6 B. 26.8 C. 28.4 D. 30.2 5-Minute Check 5

  12. A. 24.6 B. 26.8 C. 28.4 D. 30.2 5-Minute Check 5

  13. A. B. C. D. 5-Minute Check 6

  14. A. B. C. D. 5-Minute Check 6

  15. Content Standards G.C.2 Identify and describe relationships among inscribed angles, radii, and chords. G.C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Mathematical Practices 7 Look for and make use of structure. 3 Construct viable arguments and critique the reasoning of others. CCSS

  16. You found measures of interior angles of polygons. • Find measures of inscribed angles. • Find measures of angles of inscribed polygons. Then/Now

  17. inscribed angle • intercepted arc Vocabulary

  18. Concept

  19. Concept

  20. Use Inscribed Angles to Find Measures A. Find mX. Answer: Example 1

  21. Use Inscribed Angles to Find Measures A. Find mX. Answer:mX = 43 Example 1

  22. B. Use Inscribed Angles to Find Measures = 2(52) or 104 Example 1

  23. B. Use Inscribed Angles to Find Measures = 2(52) or 104 Example 1

  24. A. Find mC. A. 47 B. 54 C. 94 D. 188 Example 1

  25. A. Find mC. A. 47 B. 54 C. 94 D. 188 Example 1

  26. B. A. 47 B. 64 C. 94 D. 96 Example 1

  27. B. A. 47 B. 64 C. 94 D. 96 Example 1

  28. Concept

  29. RS R and S both intercept . Use Inscribed Angles to Find Measures ALGEBRA Find mR. mRmS Definition of congruent angles 12x – 13 = 9x + 2 Substitution x = 5 Simplify. Answer: Example 2

  30. RS R and S both intercept . Use Inscribed Angles to Find Measures ALGEBRA Find mR. mRmS Definition of congruent angles 12x – 13 = 9x + 2 Substitution x = 5 Simplify. Answer:So, mR = 12(5) – 13 or 47. Example 2

  31. ALGEBRA Find mI. A. 4 B. 25 C. 41 D. 49 Example 2

  32. ALGEBRA Find mI. A. 4 B. 25 C. 41 D. 49 Example 2

  33. Write a two-column proof. Given: Prove: ΔMNP ΔLOP Proof: Statements Reasons LO  MN2. If minor arcs are congruent, then corresponding chords are congruent. Use Inscribed Angles in Proofs 1.Given Example 3

  34. Proof: Statements Reasons M intercepts and L intercepts . 3. Definition of intercepted arc Use Inscribed Angles in Proofs M  L4. Inscribed angles of the same arc are congruent. MPN  OPL 5. Vertical angles are congruent. ΔMNP  ΔLOP 6. AAS Congruence Theorem Example 3

  35. Write a two-column proof. Given: Prove: ΔABE ΔDCE Select the appropriate reason that goes in the blank to complete the proof below. Proof: Statements Reasons AB  DC2. If minor arcs are congruent, then corresponding chords are congruent. 1.Given Example 3

  36. Proof: Statements Reasons D intercepts and A intercepts . 3. Definition of intercepted arc D  A4. Inscribed angles of the same arc are congruent. DEC  BEA 5. Vertical angles are congruent. ΔDCE  ΔABE 6. ____________________ Example 3

  37. A. SSS Congruence Theorem B. AAS Congruence Theorem C. Definition of congruent triangles D. Definition of congruent arcs Example 3

  38. A. SSS Congruence Theorem B. AAS Congruence Theorem C. Definition of congruent triangles D. Definition of congruent arcs Example 3

  39. Concept

  40. Find Angle Measures in Inscribed Triangles ALGEBRA Find mB. ΔABC is a right triangle because C inscribes a semicircle. mA + mB + mC = 180Angle Sum Theorem (x + 4) + (8x – 4) + 90 = 180 Substitution 9x + 90 = 180 Simplify. 9x = 90 Subtract 90 from each side. x = 10 Divide each side by 9. Answer: Example 4

  41. Find Angle Measures in Inscribed Triangles ALGEBRA Find mB. ΔABC is a right triangle because C inscribes a semicircle. mA + mB + mC = 180Angle Sum Theorem (x + 4) + (8x – 4) + 90 = 180 Substitution 9x + 90 = 180 Simplify. 9x = 90 Subtract 90 from each side. x = 10 Divide each side by 9. Answer:So, mB = 8(10) – 4 or 76. Example 4

  42. ALGEBRA Find mD. A. 8 B. 16 C. 22 D. 28 Example 4

  43. ALGEBRA Find mD. A. 8 B. 16 C. 22 D. 28 Example 4

  44. Concept

  45. Find Angle Measures INSIGNIAS An insignia is an emblem that signifies rank, achievement, membership, and so on. The insignia shown is a quadrilateral inscribed in a circle. Find mS and mT. Example 5

  46. Find Angle Measures Since TSUV is inscribed in a circle, opposite angles are supplementary. mS + mV = 180 mU + mT = 180 mS + 90= 180 (14x) + (8x + 4) = 180 mS = 90 22x + 4 = 180 22x = 176 x = 8 Answer: Example 5

  47. Find Angle Measures Since TSUV is inscribed in a circle, opposite angles are supplementary. mS + mV = 180 mU + mT = 180 mS + 90= 180 (14x) + (8x + 4) = 180 mS = 90 22x + 4 = 180 22x = 176 x = 8 Answer:So, mS = 90 and mT = 8(8) + 4 or 68. Example 5

  48. INSIGNIAS An insignia is an emblem that signifies rank, achievement, membership, and so on. The insignia shown is a quadrilateral inscribed in a circle. Find mN. A. 48 B. 36 C. 32 D. 28 Example 5

  49. INSIGNIAS An insignia is an emblem that signifies rank, achievement, membership, and so on. The insignia shown is a quadrilateral inscribed in a circle. Find mN. A. 48 B. 36 C. 32 D. 28 Example 5

  50. End of the Lesson

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