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Measuring Angles & Arcs

Measuring Angles & Arcs. Notes 25- Section 10.2. Essential Learnings. Students will understand and be able to identify and measure central angles, arcs and semicircles. Students will understand and be able to measure angles and arcs using degrees and radians .

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Measuring Angles & Arcs

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  1. Measuring Angles & Arcs Notes 25- Section 10.2

  2. Essential Learnings • Students will understand and be able to identify and measure central angles, arcs and semicircles. • Students will understand and be able to measure angles and arcs using degrees and radians. • Students will understand and be able to determine radian measures of angles.

  3. Essential Learnings • Students will understand and be able to identify parts of circles. • Students will understand and be able to identify and measure central angles, arcs and semicircles. • Students will understand and be able to measure angles and arcs using degrees and radians.

  4. Central Angle • Central Angle – an angle in a circle with a vertex in the center of the circle.

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

  6. Example 1 Find the value of x.

  7. Arcs • Arc – a portion of a circle defined by two endpoints. An arc is a portion of the circumference

  8. Minor Arc • Minor arc – the shortest arc connecting two endpoints on a circle. • Minor arc is named with two letters. • The measure of a minor arc is less than 180° and is equal to the measure of the central angle.

  9. Major Arc • Major arc – the longest arc connecting two endpoints on a circle. • Major arc is named with three letters. • The measure of a major arc is greater than 180° and is equal to 360 minus the measure of the minor arc with the same endpoints.

  10. Semicircle • Semicircle – is an arc with endpoints that are the diameter. • Semicircle is named with three letters. • The measure of a semicircle 180°.

  11. Example 2 AC is a diameter of circle P. Identify each arc as a major arc, minor arc, or semicircle. Minor Arcs Semicircle Major Arc

  12. Vocabulary • Congruent arcs – arcs in the same or congruent circles that have the same measure. • Adjacent arcs – arcs in a circle that have exactly one point in common.

  13. Congruent Arcs • In the same circle or in congruent circles, two minor arcs are congruent if and only if their central angles are congruent. D A 2 1 B C

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

  15. Example 3 The graph shows the breakdown of the USA budget for 2007. What are the arc measures for human resources and current military?

  16. Arc Length • The ratio of the length of an arc l to the circumference of the circle is equal to the ratio of the degree measure of the arc to 360.

  17. Example 4 In circle R, NP is the diameter. Find the length of arc PM if the radius is 5 inches.

  18. Example 5 In circle T, find the radius, round to the nearest tenth.

  19. Example 6 In circle P, m XPY = 2x and m WPX = 5x+5. Find each measure.

  20. Radians • Radians are another way to measure angles. • A Unit Circle is a circle with a radius of 1. • Circumference = 2π r • There are 2π radians in a circle.

  21. Radians • 360⁰ = 2π radians • 180⁰ = π radians

  22. Converting Between Degrees & Radians • Degrees to radians • Radians to degrees

  23. Example 7 • Convert each of the following: • 150 ⁰ • 3π/4 radians

  24. Arc Length and Area of Sector • The arc length s and area of a sector A with radius r and central angle θ (measured in radians): Angle in degrees

  25. Example 8 • Find the arc length and area of a sector with the given radius (r) and central angle (θ). • r = 5 in, θ = π/6 • r = 6 m, θ = 11π/6

  26. Assignment p. 696: 13 – 23 odd, 24, 36-41, 45-51 Degrees to Radians WS

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