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10.6 Circles

10.6 Circles. Geometry. 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.

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10.6 Circles

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  1. 10.6 Circles Geometry

  2. 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

  3. in 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 APB 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

  4. 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. 60° 60° 180°

  5. Naming Arcs • For example, the major arc associated with APB is . here on the right is a semicircle. The measure of a minor arc is defined to be the measure of its central angle. 60° 60° 180°

  6. Naming Arcs • 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° 60° 180°

  7. Naming Arcs • The measure of a major arc is defined as the difference between 360° and the measure of its associated minor arc. For example, m = 360° - 60° = 300°. The measure of the whole circle is 360°. 60° 60° 180°

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

  9. Ex. 1: Finding Measures of Arcs • Find the measure of each arc of R. Solution: is a minor arc, so m = mMRN = 80° 80°

  10. Ex. 1: Finding Measures of Arcs • Find the measure of each arc of R. Solution: is a major arc, so m = 360° – 80° = 280° 80°

  11. Ex. 1: Finding Measures of Arcs • Find the measure of each arc of R. Solution: is a semicircle, so m = 180° 80°

  12. 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. m = m + m

  13. Ex. 2: Finding Measures of Arcs • Find the measure of each arc. m = m + m = 40° + 80° = 120° 40° 80° 110°

  14. Ex. 2: Finding Measures of Arcs • Find the measure of each arc. m = m + m = 120° + 110° = 230° 40° 80° 110°

  15. Ex. 2: Finding Measures of Arcs • Find the measure of each arc. m = 360° - m = 360° - 230° = 130° 40° 80° 110°

  16. and are in the same circle and m = m = 45°. So,  Ex. 3: Identifying Congruent Arcs • Find the measures of the blue arcs. Are the arcs congruent? 45° 45°

  17. and are in congruent circles and m = m = 80°. So,  Ex. 3: Identifying Congruent Arcs • Find the measures of the blue arcs. Are the arcs congruent? 80° 80°

  18. Ex. 3: Identifying Congruent Arcs • Find the measures of the blue arcs. Are the arcs congruent? 65° m = m = 65°, but and are not arcs of the same circle or of congruent circles, so and are NOT congruent.

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