1.6 Angles and Their Measures

Transcription

1 1.6 ngles and heir Measures oal Measure 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. XML 1 ame ngles side ame the angles in the figure. olution here are three different angles. aq or aq aq or aq aq or aq 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. vertex side Mountaineers approaching asherbrum II, arakorum ange, Himalayas. a le ame the angles in the figure M H J 1.6 ngles and heir Measures 35

2 18 he measure of an angle is written in units called degrees ( ). he measure of is denoted by m. XML 2 Measure 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 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 a caq. 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. UMMY cute angle LIYI L Y HI MU Obtuse angle Visualize It! 9 small corner mark in an angle means that the angle is a right angle. Measure is between and 9. ight angle Measure is 9. Measure is between 9 and 18. traight angle Measure is hapter 1 asics of eometry

3 XML 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. XML 4 dd ngle Measures ind the measure of am. 3 olution mam ma mam 3 7 M ngle ddition ostulate ubstitute 3 for ma and for mam. dd angle measures. W he measure of am is 7. dd and ubtract ngle Measures ind the measure of a ngles and heir Measures 37

4 1.6 xercises uided ractice Vocabulary heck Match the angle with its classification.. acute. obtuse. right. straight kill heck ame the vertex and the sides of the angle. hen estimate the measure of the angle H 8. J L M lassify the angle as acute, right, obtuse, or straight. 9. ma ma ma ma 9 ractice and pplications Use the diagram at the right to answer the questions. xplain your answers. 13. Is a ca? Is a cah? 45 H xtra ractice ee p aming arts of an ngle ame the vertex and the sides of the angle X Homework H lp aming ngles Write two names for the angle. xample 1: xs xample 2: xs xample 3: xs , 3 33 xample 4: xs U 38 hapter 1 asics of eometry

5 18 areers Measuring ngles opy the angle and use a protractor to measure it to the nearest degree. xtend the sides of the angle if necessary 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 LZO.OM H 26. M L ngle ddition ostulate ind the measure of the angle. 27. m? 28. m? 29. m Q? XML 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 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 Measures 39

6 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. It is marked by the sculpture above, which depicts the prime meridian on the globe. pplication Links LZO.OM tudent Help LOO or an example of runway numbers, see the page facing p 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 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.. ind ma ind ma ind ma ind ma What is the number of the unlabeled runway? ew York ampa W ? 33 tandardized est ractice 45. Multi-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. hapter 1 asics of eometry