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10.2 Angles and Arcs

10.2 Angles and Arcs. Objectives. Recognize major arcs, minor arcs, semicircles, and central angles and their measures Find arc length. Angles and Arcs. Sum of Central s and Arcs = 360 °. Angles and Arcs. The sum of the measures of the central angles is 360 °.

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10.2 Angles and Arcs

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  1. 10.2 Angles and Arcs

  2. Objectives • Recognize major arcs, minor arcs, semicircles, and central angles and their measures • Find arc length

  3. Angles and Arcs Sum of Central s and Arcs = 360°

  4. Angles and Arcs • The sum of the measures of the central angles is 360°. • A minor arc is less then 180° and is labeled using the two endpoints. • A major arc is greater than 180° but less than 360° and is labeled using the two endpoints and another point on the arc. • A semicircle measures 180° and is labeled using the two endpoints and another point on the arc.

  5. Angles and Arcs • Theorem 10.2: In the same circle or  circles, two arcs are  if their corresponding central angles are . • Postulate 10.1: The measure of an arc formed by two adjacent arcs is the sum of the measures of the arcs.

  6. ALGEBRA Refer to . Find . Example 1a:

  7. The sum of the measures of Use the value of x to find Example 1a: Substitution Simplify. Add 2 to each side. Divide each side by 26. Given Substitution Answer: 52

  8. ALGEBRA Refer to . Find . Example 1b:

  9. form a linear pair. Example 1b: Linear pairs are supplementary. Substitution Simplify. Subtract 140 from each side. Answer: 40

  10. ALGEBRA Refer to . a. Find m b. Find m Your Turn: Answer: 65 Answer: 40

  11. In bisects and Find . Example 2a:

  12. is a minor arc, so is a semicircle. is a right angle. Example 2a: Arc Addition Postulate Substitution Subtract 90 from each side. Answer: 90

  13. In bisects and Find . Example 2b:

  14. since bisects . is a semicircle. Example 2b: Arc Addition Postulate Subtract 46 from each side. Answer: 67

  15. In bisects and Find . Example 2c:

  16. Example 2c: Vertical angles are congruent. Substitution. Substitution. Subtract 46 from each side. Substitution. Subtract 44 from each side. Answer: 316

  17. In and are diameters, and bisects Find each measure. a. b. c. Your Turn: Answer: 54 Answer: 72 Answer: 234

  18. BICYCLES This graph shows the percent of each type of bicycle sold in the United States in 2001. Find the measurement of the central angle representing each category. List them from least to greatest. Example 3a:

  19. The sum of the percents is 100% and represents the whole. Use the percents to determine what part of the whole circle each central angle contains. Answer: Example 3a:

  20. Example 3b: BICYCLES This graph shows the percent of each type of bicycle sold in the United States in 2001. Is the arc for the wedge named Youth congruent to the arc for the combined wedges named Other and Comfort?

  21. The arc for the wedge named Youth represents 26% or of the circle. The combined wedges named Other and Comfort represent . Since º, the arcs are not congruent. Example 3b: Answer: no

  22. SPEED LIMITS This graph shows the percent of U.S. states that have each speed limit on their interstate highways. Your Turn:

  23. a. Find the measurement of the central angles representing each category. List them from least to greatest. b.Is the arc for the wedge for 65 mph congruent to the combined arcs for the wedges for 55 mph and 70 mph? Answer: Your Turn: Answer: no

  24. Arc Length • Another way to measure an arc is by its length. An arc is part of a circle, so its length is part of the circumference. We use proportions to solve for the arc length, l. degree measure of arc = arc length degree measure of circumference

  25. In and . Find the length of . In and . Write a proportion to compare each part to its whole. Example 4:

  26. degree measure of arc arc length circumference degree measure of whole circle Now solve the proportion for . Multiply each side by 9 . Answer: The length of is units or about 3.14 units. Example 4: Simplify.

  27. In and . Find the length of . Answer: units or about 49.48 units Your Turn:

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