age 1 of 7 1.6 ngles and heir easures oal easure and classify angles. dd angle measures. ey Words angle sides and vertex of an angle measure of an angle degree congruent angles acute, right, obtuse, and straight angle n angle consists of two rays that have the same endpoint. he rays are the sides he endpoint is the vertex the angle. of the angle. of In the photograph at the right, the sides of a are &( and &(. he vertex of a is point. You can also write a as a. otice that the middle letter in the name of the angle is always the vertex of the angle. You can write simply a if there are no other angles that have this vertex. XL ame the angles in the figure. olution here are three different angles. aq or aq aq or aq aq or aq 1 ame ngles You should not name any of these angles as aq, because all three angles have Q as their vertex. he name aq would not distinguish one angle from the others. side vertex side ountaineers approaching asherbrum II, arakorum ange, Himalayas. ame ngles ame the angles in the figure. 1. 2. 3. H J 1.6 ngles and heir easures 35
18 age 2 of 7 he measure of an angle is written in units called degrees ( ). he measure of is denoted by m. XL 2 easure ngles Use a protractor to approximate the measure of a. olution 1 lace the center of the protractor over the vertex point. 2 lign the protractor with one side of the angle. 3 he second side of the angle crosses the protractor at the 5 mark. o, ma 5. 4 14 3 15 2 16 1 17 5 13 6 12 7 11 8 1 1 9 1 8 11 7 12 6 13 5 14 4 15 3 16 2 17 1 18 1 2 3 4 5 6 3 2 wo angles are congruent angles if they have the same measure. In the diagram below, the two angles have the same measure, so a is congruent to aq. You can write acaq. tudent Help VIUL Y dd these words to the vocabulary pages in your notebook, as shown on p. 2. ngles are classified as acute, right, obtuse, or straight, according to their measures. ngles have measures greater than and less than or equal to 18. UY cute angle LIYI L Y HI U Obtuse angle Visualize It! 9 small corner mark in an angle means that the angle is a right angle. easure is between and 9. ight angle easure is 9. easure is between 9 and 18. traight angle easure is 18. 36 hapter 1 asics of eometry
age 3 of 7 XL 3 lassify ngles lassify each angle. a. ma 13 b. ma 9 c. ma 45 olution a.a is obtuse because its measure is greater than 9. b.a is right because its measure is 9. c.a is acute because its measure is less than 9. OUL 6 Visualize It! point is in the interior of an angle if it is between points that lie on each side of the angle. ngle ddition ostulate Words If is in the interior of a, then the measure of a is the sum of the measures of a and a. ymbols If is in the interior of a, then ma ma ma. XL 4 dd ngle easures ind the measure of a. 3 olution ma ma ma 3 4 7 4 ngle ddition ostulate ubstitute 3 for ma and 4 for ma. dd angle measures. W he measure of a is 7. dd and ubtract ngle easures ind the measure of a. 4. 5. 6. 4 6 2 9 6 135 1.6 ngles and heir easures 37
age 4 of 7 1.6 xercises uided ractice Vocabulary heck atch the angle with its classification.. acute. obtuse. right. straight 1. 2. 3. 4. kill heck ame the vertex and the sides of the angle. hen estimate the measure of the angle. 5. 6. 7. H 8. L J lassify the angle as acute, right, obtuse, or straight. 9. ma 18 1. ma 34 11. ma 1 12. ma 9 ractice and pplications Use the diagram at the right to answer the questions. xplain your answers. 13. Is aca? 45 14. Is acah? 45 H xtra ractice ee p. 676. aming arts of an ngle ame the vertex and the sides of the angle. 15. 16. 17. X Homework Help aming ngles Write two names for the angle. xample 1: xs. 15 2 xample 2: xs. 21 23 xample 3: xs. 24 26, 3 33 xample 4: xs. 27 29 18. 19. 2. U 38 hapter 1 asics of eometry
18 age 5 of 7 areers easuring ngles opy the angle and use a protractor to measure it to the nearest degree. xtend the sides of the angle if necessary. 21. 22. 23. Y Z X lassifying ngles tate whether the angle appears to be acute, right, obtuse, or straight. hen estimate its measure. UVYO urveyors use a tool called a theodolite, which can measure angles to the nearest 1/36 of a degree. areer Links L Z O. O 24. 25. H 26. L ngle ddition ostulate ind the measure of the angle. 27. m? 28. m? 29. m Q? 45 6 6 12 2 16 XL lot the points (3, ), (, ), ( 2, 2) and sketch a. lassify the angle. y olution lot the points. Use a protractor to estimate the angle measure. his angle has a measure of 135. o, a is obtuse. ngles on the oordinate lane 4 14 3 15 2 16 1 17 5 13 6 12 7 11 8 1 9 1 8 11 7 12 6 13 5 14 4 15 3 16 2 17 1 18 1 2 3 4 5 6 x In xercises 3 33, use the example above as a model. lot the points and sketch a. lassify the angle. 3. (3, ), (, ), (, 3) 31. (3, ), (, ), (4, 4) 32. ( 3, ), (, ), (2, 2) 33. (, 4), (, ), (2, 2) 1.6 ngles and heir easures 39
age 6 of 7 eography eography or each city, estimate the measure of ao, where is on the longitude line, O is the orth ole, and is the city. 18 orth ole O reenwich, ngland Honolulu nchorage 15 3 LOIU he line of longitude runs through a telescope at the oyal Observatory in reenwich, ngland. pplication Links L Z O. O tudent Help LOO or an example of runway numbers, see the page facing p. 1. 12 an iego hicago 34. ew York, Y 35. ampa, L 36. hicago, IL 37. an iego, 38. Honolulu, HI 39. nchorage, 9 irport unways In xercises 4 44, use the diagram of onald eagan Washington ational irport and the information about runway numbering on the page facing page 1. n airport runway number is its bearing (the angle measured clockwise from due north) divided by 1. ecause a full circle contains 36, runway numbers range from 1 to 36. 4. ind ma1. 41. ind ma2. 42. ind ma3. 43. ind ma4. 44. What is the number of the unlabeled runway? ew York ampa W 6 3 15 1 2 18 36 3 4? 33 tandardized est ractice 45. ulti-tep roblem old a piece of paper in half three times and label it as shown. H O a. ame eight congruent acute angles. b. ame eight right angles. c. ame eight congruent obtuse angles. 4 hapter 1 asics of eometry
age 7 of 7 ixed eview aming ays ame the ray described. (Lesson 1.3) 46. ame a ray that contains. 47. ame a ray that has as an endpoint. 48. ame two rays that intersect at. etweenness raw a sketch of the three collinear points. hen write the egment ddition ostulate for the points. (Lesson 1.5) 49. Y is between X and Z. 5. Q is between and. 51. is between and. 52. is between J and L. lgebra kills One-tep quations olve the equation. (kills eview p. 672) 53. x 3 15 54. y 4 2 55. z 7 9 56. w 5 2 57. 2p 24 58. 9 3q 59. 5r 125 6. 12 6s Quiz 2 In xercises 1 3, use the photo shown at the right. (Lesson 1.4) 1. ame two lines that do not appear to intersect. 2. ame two lines that intersect at. 3. ame two lines that intersect at. Judi rown competing at the ummer Olympics. lot the points in a coordinate plane. ecide whether &* and &* are congruent. (Lesson 1.5) 4. (, ), (5, ), (2, 4), (2, 1) 5. ( 3, 2), (3, 2), (6, ), ( 6, ) lassify the angle as acute, right, obtuse, or straight. (Lesson 1.6) 6. m Z 9 7. m Y 126 8. m X 18 9. m W 35 1. mav 5 11. mau 45 Use the ngle ddition ostulate to find the measure of the angle. (Lesson 1.6) 12. ind majl. 13. ind mawxy. 14. ind mavuw. J 55 25 L Y Z 3 15 X W W U 35 95 V 1.6 ngles and heir easures 41