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Circle Geometry: Understanding Central Angles with Various Vertex Positions

This guide explores the relationships between central angles, arcs, and triangles in circles, focusing on cases where the vertex is inside, on, or outside the circle. Each case is illustrated with examples that demonstrate how to find missing angle measures using relevant formulas and concepts, including properties of right triangles and the semicircle theorem. Perfect for students looking to enhance their understanding of geometry in circles and apply these concepts effectively in problem-solving scenarios.

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Circle Geometry: Understanding Central Angles with Various Vertex Positions

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  1. Central Angle Vertex OUTSIDE circle Vertex ON circle Vertex INSIDE circle Right Triangles

  2. Time to make a Wheel of Formulas

  3. Cut out this part.

  4. Central Angle Vertex OUTSIDE circle Vertex ON circle Vertex INSIDE circle Right Triangles

  5. Case I:Vertex is ON the circle ANGLE ARC ARC ANGLE

  6. Ex. 1 Find m1. A B 1 124° C m1 = 62°

  7. Ex. 2 Find m1. 1 84° m1 = 42°

  8. Ex. 3 Find the value of x. x = 55°

  9. Ex. 4 Find the mCADif mAB = 60. mCAD= 30°

  10. Case II:Vertex is inside the circle A ARC B ANGLE D ARC C Looks like a PLUS sign!

  11. Ex. 5 Find m1. 93° A B 1 D C 113° m1 = 103°

  12. Ex. 6 Find mQT. mQT = 100° N Q 84° 92° M T

  13. Case III:Vertex is outside the circle C ANGLE small ARC A D LARGE ARC B LARGE ARC LARGE ARC small ARC ANGLE small ARC ANGLE

  14. Ex. 7 Find m1. 1 15° A D 65° B m1 = 25°

  15. Ex. 8 Find mAB. mAB = 16° A 27° 70° B

  16. Ex. 9 Find m1. 260° 1 m1 = 80°

  17. Case IV:Triangle in a semicircle semicircle (180°) semicircle DIAMETER ANGLE ANGLE = 90°

  18. Ex. 10 Find m1. AB is a diameter. m1 = 40° A 1 50° B

  19. Ex. 11 Find m1. AB is a diameter. m1 = 46° A 1 88° C B

  20. Case V:Central Angle ARC ANGLE = ARC ANGLE CENTER

  21. Ex. 12 Find mAB. mAB= 32° C 32° B A

  22. Ex. 13 Find mBC. AC is a diameter. mBC= 148° C 32° B A

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