Describe Angle Pair Relationships

Transcription

1 .5 escribe ngle Pair Relationships efore You used angle postulates to measure and classify angles. Now You will use special angle relationships to find angle measures. Why? So you can find measures in a building, as in x. 53. ey Vocabulary complementary angles supplementary angles adjacent angles linear pair vertical angles Two angles are complementary angles if the sum of their measures is 908. ach angle is the complement of the other. Two angles are supplementary angles if the sum of their measures is 808. ach angle is the supplement of the other. omplementary angles and supplementary angles can be adjacent angles or nonadjacent angles. djacent angles are two angles that share a common vertex and side, but have no common interior points. omplementary angles Supplementary angles djacent Nonadjacent djacent Nonadjacent XMP Identify complements and supplements VOI RRORS In xample, a and a share a common vertex. ut they share common interior points, so they are not adjacent angles. In the figure, name a pair of complementary R angles, a pair of supplementary angles, and a pair of adjacent angles S T ecause , and RST are complementary angles. ecause , and RST are supplementary angles. ecause and share a common vertex and side, they are adjacent. UI PRTI for xample. In the figure, name a pair of complementary angles, a pair of supplementary angles, and a pair of adjacent angles.. re and adjacent angles? re and adjacent angles? xplain escribe ngle Pair Relationships 35

2 XMP ind measures of a complement and a supplement R IRMS ngles are sometimes named with numbers. n angle measure in a diagram has a degree symbol. n angle name does not. a. iven that is a complement of and m 5 688, find m. b. iven that 3 is a supplement of 4 and m , find m 3. a. You can draw a diagram with complementary adjacent angles to illustrate the relationship. m 5 908m b. You can draw a diagram with supplementary adjacent angles to illustrate the relationship. m m XMP 3 ind angle measures R IRMS In a diagram, you can assume that a line that looks straight is straight. In xample 3,,, and lie on ]. So, is a straight angle. SPORTS When viewed from the side, the frame of a ball-return net forms a pair of supplementary angles with the ground. ind m and m. STP Use the fact that the sum of the measures of supplementary angles is 808. m m Write equation. (4x 8)8 (x ) Substitute. 5x x 5 70 ombine like terms. Subtract 0 from each side. x 5 34 ivide each side by 5. STP valuate the original expressions when x m 5 (4x 8)85 (4 p 34 8) m 5 (x )85 (34 ) c The angle measures are 448 and 368. UI PRTI for xamples and 3 3. iven that is a complement of and m 5 88, find m. 4. iven that 3 is a supplement of 4 and m , find m MN and PQR are complementary angles. ind the measures of the angles if m MN 5 (4x )8 and m PQR 5 (9x )8. 36 hapter ssentials of eometry

3 N PIRS Two adjacent angles are alinear pair if their noncommon sides are opposite rays. The angles in a linear pair are supplementary angles. Two angles are vertical angles if their sides form two pairs of opposite rays and are a linear pair. 3 and 6 are vertical angles. 4 and 5 are vertical angles. XMP 4 Identify angle pairs VOI RRORS In the diagram, one side of and one side of 3 are opposite rays. ut the angles are not a linear pair because they are not adjacent. Identify all of the linear pairs and all of the vertical angles in the figure at the right. To find vertical angles, look for angles formed by intersecting lines. c and 5 are vertical angles To find linear pairs, look for adjacent angles whose noncommon sides are opposite rays. c and 4 are a linear pair. 4 and 5 are also a linear pair. XMP 5 ind angle measures in a linear pair R Two angles form a linear pair. The measure of one angle is 5 times the measure of the other. ind the measure of each angle. RW IRMS You may find it useful to draw a diagram to represent a word problem like the one in xample 5. et x8 be the measure of one angle. The measure of the other angle is 5x. Then use the fact that the angles of a linear pair are supplementary to write an equation. x8 5x x 5 80 Write an equation. ombine like terms. 5x8 x8 x 5 30 ivide each side by 6. c The measures of the angles are 308 and 5(308) UI PRTI for xamples 4 and 5 6. o any of the numbered angles in the diagram at the right form a linear pair? Which angles are vertical angles? xplain. 7. The measure of an angle is twice the measure of its complement. ind the measure of each angle escribe ngle Pair Relationships 37

4 ONPT SUMMRY or Your Notebook Interpreting a iagram There are some things you can conclude from a diagram, and some you cannot. or example, here are some things that you can conclude from the diagram at the right: ll points shown are coplanar. Points,, and are collinear, and is between and. ] ],, and ] intersect at point. and are adjacent angles, and is a straight angle. Point lies in the interior of. In the diagram above, you cannot conclude that } > }, that >, or that is a right angle. This information must be indicated, as shown at the right..5 XRISS SI PRTI OMWOR Y 5 WOR-OUT SOUTIONS on p. WS for xs. 9,, and 47 5 STNRIZ TST PRTI xs., 6, 30, and 53 5 MUTIP RPRSNTTIONS x. 55. VOURY Sketch an example of adjacent angles that are complementary. re all complementary angles adjacent angles? xplain.. WRITIN re all linear pairs supplementary angles? re all supplementary angles linear pairs? xplain. XMP on p. 35 for xs. 3 7 INTIYIN NS Tell whether the indicated angles are adjacent. 3. and 4. WXY and XYZ 5. QM and NQM M W Z P X Y N INTIYIN NS Name a pair of complementary angles and a pair of supplementary angles. 6. P T V S U R W 7. J 38 hapter ssentials of eometry

5 XMP on p. 36 for xs. 8 6 OMPMNTRY NS and are complementary angles. iven the measure of, find m. 8. m m m m 5 58 SUPPMNTRY NS and are supplementary angles. iven the measure of, find m.. m m m m MUTIP OI The arm of a crossing gate moves 378 from vertical. ow many more degrees does the arm have to move so that it is horizontal? XMP 3 on p. 36 for xs R indm and m (8x 9)8 (4x 3)8 (7x 3)8 (x 7)8 6x8 4x8 XMP 4 on p. 37 for xs. 0 7 XMP 5 on p. 37 for xs INTIYIN N PIRS Use the diagram below. Tell whether the angles are vertical angles, a linear pair, or neither. 0. and 4. and. 3 and 5 3. and , 8, and and and and 9 8. R Two angles form a linear pair. The measure of one angle is 4 times the measure of the other angle. ind the measure of each angle. 9. RROR NYSIS escribe and correct the error made in finding the value of x. 3x8 x x8 3x x5 80 x MUTIP OI The measure of one angle is 48 greater than the measure of its complement. What are the measures of the angles? 48 and and and and 08 R ind the values ofx and y. 3. (9x 0)8 y8 7x8 3. (5y 38)8 (8x 6)8 3x8 33. y 8 (4x 00)8 (3y 30)8 (x 5)8.5 escribe ngle Pair Relationships 39

6 RSONIN Tell whether the statement isalways, sometimes, or never true. xplain your reasoning. 34. n obtuse angle has a complement. 35. straight angle has a complement. 36. n angle has a supplement. 37. The complement of an acute angle is an acute angle. 38. The supplement of an acute angle is an obtuse angle. ININ NS and are complementary. ind m and m. 39. m 5 (3x )8 40. m 5 (5x 3)8 4. m 5 (x 4)8 m 5 (x 4)8 m 5 (5x 3)8 m 5 (x 8)8 ININ NS and are supplementary. ind m and m. 4. m 5 (8x 00)8 43. m 5 (x 0)8 44. m 5 (6x 7)8 m 5 (x 50)8 m 5 (3x 5)8 m 5 (x 8)8 45. N You are given that J is a complement of RST and RST is a supplement of. et m J be x8. What is the measure of? xplain your reasoning. PROM SOVIN INTIYIN NS Tell whether the two angles shown are complementary, supplementary, or neither RITTUR The photograph shows the Rock and Roll all of ame in leveland, Ohio. Use the photograph to identify an example type of the indicated type of angle pair. 49. Supplementary angles 50. Vertical angles 5. inear pair 5. djacent angles 53. SORT RSPONS Use the photograph shown at the right. iven that and are supplementary angles, and m 5 08, explain how to find the measure of the complement of WOR-OUT SOUTIONS on p. WS 5 STNRIZ TST PRTI 5 MUTIP RPRSNTTIONS

7 54. SOWS The length of a shadow changes as the sun rises. In the diagram below, the length of } is the length of a shadow. The end of the shadow is the vertex of, which is formed by the ground and the sun s rays. escribe how the shadow and angle change as the sun rises. 55. MUTIP RPRSNTTIONS etx8 be an angle measure. et y 8 be the measure of a complement of the angle and let y 8 be the measure of a supplement of the angle. a. Writing an quation Write equations fory as a function of x, and for y as a function of x. What is the domain of each function? xplain. b. rawing a raph raph each function anddescribe its range. 56. N The sum of the measures of two complementary angles exceeds the difference of their measures by 868. ind the measure of each angle. xplain how you found the angle measures. MIX RVIW Make a table of values and graph the function. (p. 884) 57. y 5 5 x 58. y 5 3x 59. y 5 x 60. y 5x PRVIW Prepare for esson.6 in xs In each figure, name the congruent sides and congruent angles. (pp. 9, 4) 6. J QUIZ for essons.4.5 In each diagram, ] bisects. ind m and m. (p. 4). (x 0)8. 3. (8x 7)8 (0x 4)8 (6x 0)8 (3x 4)8 (9x 36)8 ind the measure of (a) the complement and (b) the supplement of. (p. 35) 4. m m m m 5 8 XTR PRTI for esson.5, p. 897 ONIN QUIZ at classzone.com 4