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10.2 Finding Arc Measures

10.2 Finding Arc Measures. 2/24/2010. Objectives/Assignment. Use properties of arcs of circles, as applied. In a plane, an angle whose vertex is the center of a circle is a central angle of the circle.

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10.2 Finding Arc Measures

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  1. 10.2 Finding Arc Measures 2/24/2010

  2. Objectives/Assignment • Use properties of arcs of circles, as applied.

  3. 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 are the points of P Using Arcs of Circles

  4. 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 • Minor Arc: < 180˚ • Major Arc: > 180˚

  5. Naming Arcs • Arcs are named by their endpoints. • Major arcs and semicircles are named by their endpoints and by a point on the arc. (3 Letters). • The minor arc is named • The major arc is named

  6. Naming Arcs • Name the semicircle. • The measure of a minor arc is defined to be the measure of its central angle. 60° 60°

  7. 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°

  8. Naming Arcs • The measure of the whole circle is 360°. • The measure of a major arc is defined as the difference between 360° and the measure of its associated minor arc. • Find m . 60° 60° 180°

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

  10. 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°

  11. 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°

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

  13. 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

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

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

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

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