Measuring Angles & Arcs

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