Essential Question How can you measure and classify an angle?

Transcription

1 easuring and onstructing ngles ssential Question ow can you measure and classify an angle? easuring and lassifying ngles Work with a partner. ind the degree measure of each of the following angles. lassify each angle as acute, right, or obtuse O 1 0 a. O b. O c. O d. O e. O f. O g. O h. O rawing a egular olygon Work with a partner. a. Use a ruler and protractor to draw the triangular pattern shown at the right. b. ut out the pattern and use it to draw three regular hexagons, as shown below. 1 TTI TO IIO To be proficient in math, you need to calculate and measure accurately and efficiently. c. The sum of the angle measures of a polygon with n sides is equal to 1(n 2). o the angle measures of your hexagons agree with this rule? xplain. d. artition your hexagons into smaller polygons, as shown below. or each hexagon, find the sum of the angle measures of the smaller polygons. oes each sum equal the sum of the angle measures of a hexagon? xplain. ommunicate Your nswer 3. ow can you measure and classify an angle? ection 1.5 easuring and onstructing ngles 37

2 1.5 esson What You Will earn ame angles. easure and classify angles. ore Vocabul Vocabulary larry angle, p. 38 vertex, p. 38 sides of an angle, p. 38 interior of an angle, p. 38 exterior of an angle, p. 38 measure of an angle, p. 39 acute angle, p. 39 right angle, p. 39 obtuse angle, p. 39 straight angle, p. 39 congruent angles, p. angle bisector, p. 42 Identify congruent angles. Use the ngle ddition ostulate to find angle measures. isect angles. aming ngles n angle is a set of points consisting of two different rays that have the same endpoint, called the vertex. The rays are the sides of the angle. vertex 1 You can name an angle in several different ways. Use its vertex, such as. Use a point on each ray and the vertex, such as or. Use a number, such as 1. revious protractor degrees sides The region that contains all the points between the sides of the angle is the interior of the angle. The region that contains all the points outside the angle is the exterior of the angle. exterior interior aming ngles OO O When a point is the vertex of more than one angle, you cannot use the vertex alone to name the angle. lighthouse keeper measures the angles formed by the lighthouse at point and three boats. ame three angles shown in the diagram. J ligh lighthouse htho o ouse OUTIO J or J or J or J onitoring rogress elp in nglish and panish at igideasath.com Write three names for the angle Q 38 hapter 1 hs_geo_pe_05.indd X 1 Z Y asics of eometry 1/19/15 8:22

3 OO O ost protractors have an inner and an outer scale. When measuring, make sure you are using the correct scale. easuring and lassifying ngles protractor helps you approximate the measure of an angle. The measure is usually given in degrees. ostulate ostulate 1.3 rotractor ostulate onsider O and a point on one side of O. The rays of the form O can be matched one to one with the real numbers from 0 to 1. The measure of O, which can be written as m O, is equal to the absolute value of the difference between the real numbers matched with O and O on a protractor. 1 1 O You can classify angles according to their measures. ore oncept Types of ngles acute angle right angle obtuse angle straight angle easures greater than 0 and less than easures easures greater than and less than 1 easures 1 ind the measure of each angle. Then classify each angle. a. b. J c. easuring and lassifying ngles OUTIO a. lines up with 0 on the outer scale of the protractor. passes J through 125 on the outer scale. o, m = 125. It is an obtuse angle. b. J lines up with 0 on the inner scale of the protractor. passes through. o, m J =. It is a right angle. c. passes through. passes through 55 on the inner scale. o, m = 55 = 35. It is an acute angle. onitoring rogress elp in nglish and panish at igideasath.com Use the diagram in xample 2 to find the angle measure. Then classify the angle. 4. J ection 1.5 easuring and onstructing ngles 39

4 Identifying ongruent ngles You can use a compass and straightedge to construct an angle that has the same measure as a given angle. opying an ngle Use a compass and straightedge to construct an angle that has the same measure as. In this construction, the center of an arc is the point where the compass point rests. The radius of an arc is the distance from the center of the arc to a point on the arc drawn by the compass. OUTIO tep 1 tep 2 tep 3 tep 4 raw a segment raw an angle such as, as shown. Then draw a segment. abel a point on the segment. raw arcs raw an arc with center. Using the same radius, draw an arc with center. raw an arc abel,, and. raw an arc with radius and center. abel the intersection. raw a ray raw.. Two angles are congruent angles when they have the same measure. In the construction above, and are congruent angles. o, m = m The measure of angle is equal to the measure of angle. and. ngle is congruent to angle. Identifying ongruent ngles a. Identify the congruent angles labeled in the quilt design. b. m =. What is m? OUTIO a. There are two pairs of congruent angles: I In diagrams, matching arcs indicate congruent angles. When there is more than one pair of congruent angles, use multiple arcs. and. b. ecause, m = m. o, m =. onitoring rogress elp in nglish and panish at igideasath.com 7. Without measuring, is in xample 3? xplain your reasoning. Use a protractor to verify your answer. hapter 1 asics of eometry

5 Using the ngle ddition ostulate ostulate ostulate 1.4 ngle ddition ostulate Words If is in the interior of T, then the measure of T is equal to the sum of the measures of and T. ymbols If is in the interior of T, then m T = m + m T. m T m m T T iven that m = 145, find m and m. inding ngle easures (2x + ) OUTIO tep 1 Write and solve an equation to find the value of x. (4x 3) m = m + m ngle ddition ostulate 145 = (2x + ) + (4x 3) ubstitute angle measures. 145 = 6x + 7 ombine like terms. 138 = 6x ubtract 7 from each side. 23 = x ivide each side by 6. tep 2 valuate the given expressions when x = 23. m = (2x + ) = ( ) = 56 m = (4x 3) = (4 23 3) = 89 o, m = 56, and m = 89. onitoring rogress elp in nglish and panish at igideasath.com ind the indicated angle measures. 8. iven that is a straight angle, 9. iven that is a right angle, find m and m. find m and m. (x 5) (4x + 3) (2x + 2) (x + 1) ection 1.5 easuring and onstructing ngles 41

6 isecting ngles n angle bisector is a ray that divides an angle into two angles that are congruent. In the figure, YW bisects XYZ, so XYW ZYW. Y X W You can use a compass and straightedge to bisect an angle. Z isecting an ngle onstruct an angle bisector of with a compass and straightedge. OUTIO tep 1 tep 2 tep 3 in cm raw an arc raw an angle such as, as shown. lace the compass at. raw an arc that intersects both sides of the angle. abel the intersections and. raw arcs lace the compass at. raw an arc. Then place the compass point at. Using the same radius, draw another arc. raw a ray abel the intersection. Use a straightedge to draw a ray through and. bisects. Using a isector to ind ngle easures Q bisects Q, and m Q = 24. ind m Q. OUTIO tep 1 raw a diagram. Q 24 tep 2 ecause Q bisects Q, m Q = m Q. o, m Q = 24. Use the ngle ddition ostulate to find m Q. m Q = m Q + m Q o, m Q = 48. onitoring rogress ngle ddition ostulate = ubstitute angle measures. = 48 dd.. ngle is a straight angle, and elp in nglish and panish at igideasath.com Q bisects. raw and Q. Use arcs to mark the congruent angles in your diagram. ind the angle measures of these congruent angles. 42 hapter 1 asics of eometry

7 xercises ynamic olutions available at igideasath.com Vocabulary and ore oncept heck 1. OT T T Two angles are angles when they have the same measure. 2. WI O O T O? Which angle name does not belong with the other three? xplain your reasoning onitoring rogress and odeling with athematics In xercises 3 6, write three names for the angle. (ee xample 1.) In xercises 9 12, find the angle measure. Then classify the angle. (ee xample 2.) O In xercises 7 and 8, name three different angles in the diagram. (ee xample 1.) 7. J 1 8. Q J 8 9. m O. m O 11. m O 12. m O O YI In xercises 13 and 14, describe and correct the error in finding the angle measure. Use the diagram from xercises m O = m O = 65 ection 1.5 easuring and onstructing ngles 43

8 OTUTIO In xercises 15 and 16, use a compass and straightedge to copy the angle In xercises 25, find the indicated angle measures. ynamic (ee olutions xample available 4.) at igideasath.com 25. m = 95. ind m and m. (2x + 23) In xercises 17, m = 34 and m = 112. (ee xample 3.) (9x 5) 17. Identify the angles congruent to. 18. Identify the angles congruent to. 19. ind m.. ind m. In xercises 21 24, find the indicated angle measure. 21. ind m. 22. ind m m T = 114. ind m V m XYZ = 117. ind m XYW and m WYZ. Y Z ( x + 65) (6x + 44) X W 27. is a straight angle. ind m and m. ( 16x + 13) ( x + 23) 28. is a straight angle. ind m X and m X. X (14x + ) (x + 8) 29. ind m Q and m TQ. V (15x 43) Q 72 T (8x + 18) T 24. is a straight angle. ind m.. ind m and m. (x + 21) 79 13x 44 hapter 1 asics of eometry

9 OTUTIO In xercises 31 and 32, copy the angle. Then construct the angle bisector with a compass and straightedge YZI TIOI The map shows the intersections of three roads. alcom Way intersects ydney treet at an angle of 162. ark oad intersects ydney treet at an angle of 87. ind the angle at which alcom Way intersects ark oad. In xercises 33 36, Q bisects Q. Use the diagram and the given angle measure to find the indicated angle measures. (ee xample 5.) 33. m Q = 63. ind m Q and m Q. 34. m Q = 71. ind m Q and m Q. Q ydney treet alcom Way 43. YZI TIOI In the sculpture shown in the photograph, the measure of is 76 and the measure of is 36. What is the measure of? ark oad 35. m Q = 124. ind m Q and m Q. 36. m Q = 119. ind m Q and m Q. In xercises 37, bisects. ind m, m, and m. 37. (6x + 14) (3x + 29) ( 4x + 33) (2x + 81) (3x + 6) (7x 18) (8x + 35) (11x + 23) 41. WITI xplain how to find m when you are given m and m. UI TUTU In xercises 44 46, use the diagram of the roof truss. 44. In the roof truss, bisects and, m = 112, and. ind the measure of each angle. a. m b. m c. m d. m 45. In the roof truss, is a straight angle and bisects. ind m and m. 46. ame an example of each of the four types of angles according to their measures in the diagram. ection 1.5 easuring and onstructing ngles 45

10 47. TTI OTIO In, X is in the interior of the angle, m X is 12 more than 4 times m X, and m = 92. a. raw a diagram to represent the situation. b. Write and solve an equation to find m X and m X. 48. TOUT OVOI The angle between the minute hand and the hour hand of a clock is. What time is it? Justify your answer. 49. TT OI lassify the angles that result from bisecting each type of angle. a. acute angle b. right angle c. obtuse angle d. straight angle. TT OI lassify the angles that result from drawing a ray in the interior of each type of angle. Include all possibilities and explain your reasoning. a. acute angle b. right angle c. obtuse angle d. straight angle 51. ITI TII The ray from the origin through (4, 0) forms one side of an angle. Use the numbers below as x- and y-coordinates to create each type of angle in a coordinate plane a. acute angle b. right angle c. obtuse angle d. straight angle 53. ITI TII Two acute angles are added together. What type(s) of angle(s) do they form? xplain your reasoning. 54. OW O YOU IT? Use the diagram. X W Y 46 a. Is it possible for XYZ to be a straight angle? xplain your reasoning. b. What can you change in the diagram so that XYZ is a straight angle? 55. WITI xplain the process of bisecting an angle in your own words. ompare it to bisecting a segment. 56. YZI TIOI Q bisects T, bisects Q, and V bisects. The measure of V is 17. ind m TQ. xplain. 57. TT OI bubble level is a tool used to determine whether a surface is horizontal, like the top of a picture frame. If the bubble is not exactly in the middle when the level is placed on the surface, then the surface is not horizontal. What is the most realistic type of angle formed by the level and a horizontal line when the bubble is not in the middle? xplain your reasoning. V Z 52. I UT Your friend claims it is possible for a straight angle to consist of two obtuse angles. Is your friend correct? xplain your reasoning. aintaining athematical roficiency olve the equation. (kills eview andbook) 58. x + 67 = x + 58 = x = x = 1 eviewing what you learned in previous grades and lessons 62. (6x + 7) + (13x + 21) = (3x + 15) + (4x 9) = 64. (11x 25) + (24x + ) = 65. (14x 18) + (5x + 8) = 1 46 hapter 1 asics of eometry