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Central Angles

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This resource covers the concepts of central angles and inscribed angles within circles. A central angle, with its vertex at the center, measures the same as the corresponding arc. The arc can be classified as major or minor based on its measurement in degrees. Semicircles divide circles into two equal parts at 180 degrees. Further, inscribed angles, whose vertex lies on the circle itself, measure half of the intercepted arc's angle. Use this guide to grasp key properties and solve related geometry problems.

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Central Angles

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  1. Central Angles

  2. Central Angle : An Angle whose vertex is at the center of the circle ACB AB A Major Arc Minor Arc More than 180° Less than 180° P To name: use 3 letters C To name: use 2 letters B <APB is a Central Angle

  3. EDF Semicircle: An Arc that equals 180° To name: use 3 letters E D P F EF is a diameter, so every diameter divides the circle in half, which divides it into arcs of 180°

  4. THINGS TO KNOW AND REMEMBER ALWAYS A circle has 360 degrees A semicircle has 180 degrees Vertical Angles are Equal Linear Pairs are Supplementary

  5. Vertical Angles are Equal

  6. Linear Pairs are Supplementary http://www.mathopenref.com/linearpair.html 120° 60°

  7. measure of an arc = measure of central angle m AB m ACB m AE A E 96 Q = 96° B C = 264° = 84°

  8. Arc Addition Postulate m ABC = m AB + m BC A C B

  9. m DAB = Tell me the measure of the following arcs. 240 D A 140 260 m BCA = R 40 100 80 C B

  10. CONGRUENT ARCS Congruent Arcs have the same measure and MUST come from the same circle or from congruent circles. C B D 45 45 110 A

  11. Classwork • Page 193 #9-18 You have 15 minutes.

  12. Inscribed Angle: An angle whose vertex is on the circle and whose sides are chords of the circle INTERCEPTEDARC INSCRIBEDANGLE

  13. YES; CL C T O L Determine whether each angle is an inscribed angle. Name the intercepted arc for the angle. 1.

  14. NO; QVR Determine whether each angle is an inscribed angle. Name the intercepted arc for the angle. 2. Q V K R S

  15. To find the measure of an inscribed angle… 160° 80°

  16. http://www.geogebra.org/en/upload/files/english/Guy/Circles_and_angles/Inscribed_Anlge.htmlhttp://www.geogebra.org/en/upload/files/english/Guy/Circles_and_angles/Inscribed_Anlge.html

  17. What do we call this type of angle? What is the value of x? What do we call this type of angle? How do we solve for y? The measure of the inscribed angle is HALF the measure of the inscribed arc!! 120 x y

  18. http://www.geogebra.org/en/upload/files/english/Guy/Circles_and_angles/Inscribed_angle_practice.htmlhttp://www.geogebra.org/en/upload/files/english/Guy/Circles_and_angles/Inscribed_angle_practice.html

  19. J K Q S M Examples 3. If m JK = 80, find m<JMK. 40  4. If m<MKS = 56, find m MS. 112 

  20. If two inscribed angles intercept the same arc, then they are congruent. 72

  21. http://www.geogebra.org/en/upload/files/english/Guy/Circles_and_angles/Inscribed_angle_practice.htmlhttp://www.geogebra.org/en/upload/files/english/Guy/Circles_and_angles/Inscribed_angle_practice.html

  22. Q D A J T B U Example 5 In J, m<A= 5x and m<B = 2x + 9. Find the value of x. m<A = m<B 5x = 2x+9 x = 3

  23. Classwork: • Page 193 #9-23 • Page 207 #1-15

  24. Whatever is left is homework

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