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10-2 Angles and Arcs

10-2 Angles and Arcs. Central Angle. A central angle is an angle whose vertex is at the center of a circle. Sum of Central Angles. The sum of the measures of the central angles of a circle with no interior points in common is 360. Arc. An arc is an unbroken part of a circle. Minor Arc.

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10-2 Angles and Arcs

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  1. 10-2 Angles and Arcs

  2. Central Angle • A central angle is an angle whose vertex is at the center of a circle.

  3. Sum of Central Angles • The sum of the measures of the central angles of a circle with no interior points in common is 360.

  4. Arc • An arc is an unbroken part of a circle.

  5. Minor Arc • Part of a circle that measures less than 180°.

  6. Semicircle • An arc whose endpoints are the endpoints of a diameter of the circle.

  7. Major Arc • Part of a circle that measures between 180° and 360°.

  8. Definition of Arc Measure • The measure of a minor arc is the measure of its central angle. The measure of a major arc is 360 minus the measure of its central angle. The measure of a semicircle is 180.

  9. Naming Arcs • Arcs are named by their endpoints. For example, the minor arc associated with APB above is . Major arcs and semicircles are named by their endpoints and by a point on the arc.

  10. In a plane, an angle whose vertex is the center of a circle is a central angle of the circle. If the measure of a central angle, APB is less than 180°, then A and B and the points of P Using Arcs of Circles

  11. The interior of APB form a minor arc of the circle. The points A and B and the points of P in the exterior of ACB form a major arc of the circle. If the endpoints of an arc are the endpoints of a diameter, then the arc is a semicircle. Using Arcs of Circles

  12. Naming Arcs 60° • For example, the major arc associated with APB is . The measure of a minor arc is defined to be the measure of its central angle.

  13. Naming Arcs 60° • For instance, m = mGHF = 60°. • m is read “the measure of arc GF.” You can write the measure of an arc next to the arc. The measure of a semicircle is always 180°. 60° 180°

  14. Ex. 1: Finding Measures of Arcs • Find the measure of each arc of R. 80°

  15. Adjacent Arcs • Adjacent arcs are arcs of a circle that have exactly one point in common.

  16. Note: • Two arcs of the same circle are adjacent if they intersect at exactly one point. You can add the measures of adjacent areas. • Postulate 26—Arc Addition Postulate. The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.

  17. Arc Addition Postulate • The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.

  18. Ex. 2: Finding Measures of Arcs • Find the measure of each arc. 40° 80° 110°

  19. Arc Length • A portion of the circumference of a circle.

  20. Arc Length Formula • Arc length AB = mAB • 2лr 360° 80

  21. Find the arc length of HE and FE. 4 in 75 110

  22. Concentric Circles • Concentric circles are circles that have a common center. • Concentric circles lie in the same plane and have the same center, but have different radii. • All circles are similar circles.

  23. Congruent Circles • Circles that have the same radius are congruent circles.

  24. Congruent Arcs • If two arcs of one circle have the same measure, then they are congruent arcs. • Congruent arcs also have the same arc length.

  25. Assignment page 710 • Class work 1-23 (turn in) • Homework 26-41

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