Measure and classify angles. Identify and use congruent angles and the bisector of an angle. big is a degree? One of the first references to the


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1 ngle Measure Vocabulary degree ray opposite rays angle sides vertex interior exterior right angle acute angle obtuse angle angle bisector tudy ip eading Math Opposite rays are also known as a straight angle. Its measure is 80. Unless otherwise specified, the term angle in this book means a nonstraight angle. Measure and classify angles. Identify and use congruent angles and the bisector of an angle. big is a degree? One of the first references to the 360 measure now known as a degree came from astronomer laudius tolemy. He based his observations of the solar system on a unit that resulted from dividing the circumference, or the distance around, a circle into 360 parts. his later became known as a degree. In this lesson, you will learn to measure angles in degrees. MEUE NGLE ray is part of a line. It has one endpoint and extends indefinitely in one direction. ays are named stating the endpoint first and then any other point on the ray. he figure at the right shows ray E, which can be symbolized as E. his ray could also be named as EG, but not as E because is not the endpoint of the ray. If you choose a point on a line, that point determines exactly two rays called opposite. Line m, shown below, is separated into two opposite rays, and rays. oint is the common endpoint of those rays. and are collinear rays. m E = 360 of a turn around a circle G n angle is formed by two noncollinear rays that have a common endpoint. he rays are called sides of the angle. he common endpoint is the vertex. ngle Words ymbols n angle is formed by two noncollinear rays that have a common endpoint. 4 Model vertex side 4 side n angle divides a plane into three distinct parts. oints,, and E lie on the angle. oints and lie in the interior of the angle. oints and G lie in the exterior of the angle. E G Lesson 4 ngle Measure 9
2 tudy ip Naming ngles ou can name an angle by a single letter only when there is one angle shown at that vertex. Example ngles and heir arts a. Name all angles that have W as a vertex.,, 3, W, WV b. Name the sides of. W and W are the sides of. W 3 V 5 4 c. Write another name for W. 4,, and W are other names for W. o measure an angle, you can use a protractor. ngle is a 65 degree (65 ) angle. We say that the degree measure of is 65, or simply m 65. he protractor has two scales running from 0 to 80 degrees in opposite directions ince is aligned with the 0 on the outer scale, use the outer scale to find that intersects the scale at 65 degrees. lign the 0 on either side of the scale with one side of the angle. ngles can be classified by their measures lace the center point of the protractor on the vertex. tudy ip lassifying ngles he corner of a piece of paper is a right angle. Use the corner to determine if an angle s measure is greater than 90 or less than 90. Name lassify ngles Measure m 90 m m 90 Model right angle his symbol means a 90 angle. acute angle obtuse angle Example Measure and lassify ngles Measure each angle named and classify it as right, acute, or obtuse. a. M Use a protractor to find that m M , so M is an acute angle. b. M M is marked with a right angle symbol, so measuring is not necessary; m M 90. c. M Use a protractor to find that m M 0. M is an obtuse angle. M 30 hapter oints, Lines, lanes, and ngles
3 ONGUEN NGLE Just as segments that have the same measure are congruent, angles that have the same measure are congruent. Words ngles that have the same Model measure are congruent angles. rcs on the figure also indicate which angles are congruent. N 5 ymbols NM M ongruent ngles M 5 ou can construct an angle congruent to a given angle without knowing the measure of the angle. opy an ngle raw an angle like on your paper. Use a straightedge to draw a ray on your paper. Label its endpoint. lace the tip of the compass at point and draw a large arc that intersects both sides of. Label the points of intersection and. 3 Using the same compass setting, put the compass at and draw a large arc that intersects the ray. Label the point of intersection. 4 lace the point of your compass on and adjust so that the pencil tip is on. 5 Without changing the setting, place the compass at and draw an arc to intersect the larger arc you drew in tep 3. Label the point of intersection U. 6 Use a straightedge to draw U. U U Lesson 4 ngle Measure 3
4 tudy ip hecking olutions heck that you have computed the value of x correctly by substituting the value into the expression for. If you don t get the same measure as, you have made an error. Example 3 Use lgebra to ind ngle Measures GENING trellis is often used to provide a frame for vining plants. ome of the angles formed by the slats of the trellis are congruent angles. In the figure,. If m 6x and m 8x 4, find the actual measurements of and. m m 6x 8x 4 6x 6 8x 6 x Given efinition of congruent angles ubstitution dd 4 to each side. ubtract 6x from each side. 8 x ivide each side by. (6x ) (8x 4) Use the value of x to find the measure of one angle. m 6x Given m 6(8) x 8 m 48 or 50 implify. ince m m, m 50. oth and measure 50. tudy ip dding ngle Measures Just as with segments, when a line divides an angle into smaller angles, the sum of the measures of the smaller angles equals the measure of the largest angle. o in the figure, m m m. isect an ngle Make a Model raw any on patty paper or tracing paper. old the paper through point so that and are aligned together. Open the paper and label a point on the crease in the interior of as point W. nalyze the Model. What seems to be true about W and W?. Measure, W, and W. 3. ou learned about a segment bisector in Lesson 3. Write a sentence to explain the term angle bisector. ray that divides an angle into two congruent angles is called an. If angle bisector is the angle bisector of, then point lies in the interior of and. W 3 hapter oints, Lines, lanes, and ngles ed Habegger/Grant Heilman hotography ou can construct the angle bisector of any angle without knowing the measure of the angle.
5 isect an ngle /3/003 3:8 M brian_batch GEO L raw an angle on With the compass at 3 your paper. Label the point, draw an arc vertex as. ut your in the interior of the compass at point angle. and draw a large arc that intersects both sides of. Label the points of intersection and. Keeping the same compass setting, place the compass at point and draw an 06 arc that intersects the arc drawn in tep. 4 Label the point of intersection. raw. is the bisector of. hus, m m and. oncept heck Guided ractice. etermine whether all right angles are congruent.. OEN ENE raw and label a figure to show that bisects and that bisects. Use a protractor to measure each angle. 3. Write a statement about the measures of congruent angles and. or Exercises 4 and 5, use the figure at the right. 4. Name the vertex of. 5. Name the sides of Write another name for. 4 3 Measure each angle and classify as right, acute, or obtuse. 7. W 8. W W pplication LGE In the figure, and are opposite rays, and bisects. 9. If m 6x 5 and m 7x, find m. 0. ind m if m a and m a 8.. OIGMI he art of origami involves folding paper at different angles to create designs and threedimensional figures. One of the folds in origami involves folding a strip of paper so that the lower edge of the strip forms a right angle with itself. Identify each numbered angle as right, acute, or obtuse. 3 Lesson 4 ngle Measure 33
6 ractice and pply or Exercises ee Examples 3 Extra ractice ee page 755. Name the vertex of each angle Name the sides of each angle E G 7 H Write another name for each angle E Name a point in the interior of G. 5. Name an angle with vertex that appears to be acute. 6. Name a pair of angles that share exactly one point. 7. If bisects E and m E 60, find m 5 and m 6. Measure each angle and classify it as right, acute, or obtuse E 3. E E E LGE In the figure, and are opposite rays. U bisects W, and bisects W. 34. If m U 8p 0 and m UW 0p 0, find m U. 35. If m 5x 0 and m 8x 3, find m. 36. If m y and m W 6y 4, find y. 37. If m W 8 and m U 4r 5, find r. 38. If m W (b 7) and m U 9b, find m UW. 39. If W is a right angle and m U 3a 7, find a. W U 40. OG KING dog is tracking when it is following the scent trail left by a human being or other animal that has passed along a certain route. One of the training exercises for these dogs is a tracking trail. he one shown is called an acute tracking trail. Explain why it might be called this. wind direction tart food drop article 34 hapter oints, Lines, lanes, and ngles 4. LNGUGE he words obtuse and acute have other meanings in the English language. Look these words up and write how the everyday meaning relates to the mathematical meaning.
7 4. EN LOK attern blocks can be arranged to fit in a circular pattern without leaving spaces. emember that the measurement around a full circle is 360. etermine the angle measure of the numbered angles shown below HI ripple tank can be used to study the behavior of waves in two dimensions. s a wave strikes a barrier, it is reflected. he angle of incidence and the angle of reflection are congruent. In the diagram at the right, if m I 6, find the angle of reflection and m I. I angle of incidence N angle of reflection barrier hysics ripple tank is a large glassbottomed tank of water. light is placed above the water, and a white sheet of paper is placed below the tank. ecause rays of light undergo bending as they pass through the troughs and crests of the water, there is a pattern of light and dark spots on the white sheet of paper. hese model the wave. 44. IIL HINKING How would you compare the size of and? Explain. IIL HINKING or Exercises 45 48, use the following information. Each figure below shows noncollinear rays with a common endpoint. rays 3 rays 4 rays 5 rays 6 rays tandardized est ractice 45. ount the number of angles in each figure. 46. escribe the pattern between the number of rays and the number of angles. 47. Make a conjecture of the number of angles that are formed by 7 noncollinear rays and by 0 noncollinear rays. 48. Write a formula for the number of angles formed by n noncollinear rays with a common endpoint. 49. WIING IN MH nswer the question that was posed at the beginning of the lesson. How big is a degree? Include the following in your answer: how to find degree measure with a protractor, and drawings of several angles and their degree measures. 50. If bisects, which of the following are true? m m m m m m all of these 5. LGE olve 5n 4 7(n ) n. 0 no solution all numbers Lesson 4 ngle Measure 35 (l)erich chrempp/hoto esearchers, (r)aron Haupt
8 Maintain our kills Mixed eview ind the distance between each pair of points. hen find the coordinates of the midpoint of the line segment between the points. (Lesson 3) 5. (, 3), (5, 7) 53. (, 0), (6, 4) 54. E( 3, ), (5, 8) ind the measurement of each segment. (Lesson ) 55. W 56. ft ft W 3.7 mm 5. mm 57. ind if lies between and, 6x 5, x 7, and. (Lesson ) efer to the figure at the right. (Lesson ) 58. How many planes are shown? 59. Name three collinear points. 60. Name a point coplanar with J, H, and. L K J G H Getting eady for the Next Lesson EEUIIE KILL olve each equation. (o review solving equations, see pages 737 and 738.) 6. 4x (6x 0) k y 90 7y t (80 t) (6m 8) (3m 0) (7n 9) (5n 45) 80 ractice uiz Lessons 3 and 4 ind the coordinates of the midpoint of each segment. hen find the distance between the endpoints. (Lesson 3). y. y 3. ( 4, 3) O x (3, ) (6, 4) O x (, 8) ( 0, 0) E(0, 0) In the figure, and are opposite rays. Given the following conditions, find the value of a and the measure of the indicated angle. (Lesson 4) 4. m 3a 4, m a 5, m ; m 5. m a 0, m 4a, m 9; m 36 hapter oints, Lines, lanes, and ngles
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