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Answers to Think and Discuss THINK AND DISCUSS 1. All rt. measure 90°, so the sum of the measures of any 2 rt. is 180°. Therefore any 2 rt. are supp. 1. Explain why any two right angles are supplementary. 2. Is it possible for a pair of vertical angles to also be adjacent? Explain. 3. GET ORGANIZED Copy and complete the graphic organizer below. In each box, draw a diagram and write a definition of the given angle pair. «iiÌ>ÀÞ >}ià *>ÀÃÊv }ià `>ViÌÊ>}ià i>ÀÊ«>À 6iÀÌV>Ê>}ià 1-4 2. Vert. cannot be adj. because the def. of vert. states that they are nonadj. formed by intersecting lines. -Õ««iiÌ>ÀÞ >}ià 3. See p. A2. Exercises 1-4 Exercises KEYWORD: MG7 1-4 KEYWORD: MG7 Parent GUIDED PRACTICE Assignment Guide Vocabulary Apply the vocabulary from this lesson to answer each question. (90 - x)°; 1. (180 - x)° Assign Guided Practice exercises as necessary. An angle measures x°. What is the measure of its complement? What is the measure of its supplement? BC 2. ∠ABC and ∠CBD are adjacent angles. Which side do the angles have in common? SEE EXAMPLE 1 p. 28 Tell whether the angles are only adjacent, adjacent and form a linear pair, or not adjacent. 3. ∠1 and ∠2 adj.; 4. ∠1 and ∠3 not adj. lin. pair 5. ∠2 and ∠4 6. ∠2 and ∠3 only adj. not adj. SEE EXAMPLE 2 9. supplement of ∠B SEE EXAMPLE 3 p. 29 SEE EXAMPLE 4 p. 30 Ó Î { £ If you finished Examples 1–5 Basic 14–22, 24–30, 33, 34, 37–43, 47–55 Average 14–44, 47–55 Advanced 14–55 Find the measure of each of the following. 7. supplement of ∠A 98.8° p. 29 (185 - 6x)° 8. complement of ∠A 8.8° ÈÝÊÊx®Â 10. complement of ∠B (95 - 6x)° n£°Ó If you finished Examples 1–3 Basic 14–22, 25, 27–30, 34 Average 14–22, 25, 27–35 Advanced 14–22, 25, 27–35, 37 11. Multi-Step An angle’s measure is 6 degrees more than 3 times the measure of its complement. Find the measure of the angle. 69° Homework Quick Check Quickly check key concepts. Exercises: 14, 18, 22, 24, 28, 30, 34 12. Landscaping A sprinkler swings back and forth between A and B in such a way that ∠1 ∠2. ∠1 and ∠3 are complementary, and ∠2 and ∠4 are complementary. If m∠1 = 47.5°, find m∠2, m∠3, and m∠4. Î { £ Ó m∠2 = 47.5°; m∠3 = m∠4 = 42.5° SEE EXAMPLE p. 30 5 13. Name each pair of vertical angles. ∠ABE, ∠CBD; ∠ABC, ∠EBD 1- 4 Pairs of Angles ge07se_c01_0028_0033.indd 31 31 8/29/06 9:49:56 AM KEYWORD: MG7 Resources Lesson 1-4 31 Exercise 33 involves using algebra to find measures of complementary angles, congruent angles, and linear pairs. This exercise prepares students for the Multi-Step Test Prep on page 34. PRACTICE AND PROBLEM SOLVING 14–17 18–21 22 23 24 Answers 33a. 1 2 3 4 5 19. complement of ∠A 33.6° 20. supplement of ∠B (184 - 2x)° 21. complement of ∠B (94 - 2x)° 23. Art In the stained glass pattern, ∠1 ∠2. ∠1 and ∠3 are complementary, and ∠2 and ∠4 are complementary. If m∠1 = 22.3°, find m∠2, m∠3, and m∠4. H K A J 18. supplement of ∠A 123.6° 22. Multi-Step An angle’s measure is 3 times the measure of its complement. Find the measure of the angle and the measure of its complement. 67.5°; 22.5° Application Practice p. S28 b. J K 24. Name the pairs of vertical angles. A ∠PTU, ∠VTR; ∠UTQ, ∠STV ; ∠QTR, ∠PTS; ∠PTQ, ∠STR; ∠UTR, ∠PTV ; ∠QTV, ∠UTS H 1 2 + 1 * 4 3 m∠2 = 22.3°; m∠3 = m∠4 = 67.7° H c. £ { Given m∠A = 56.4° and m∠B = (2x - 4)°, find the measure of each of the following. Skills Practice p. S4 K Ó Î 14. ∠1 and ∠4 adj.; 15. ∠2 and ∠3 adj.; lin. pair lin. pair 16. ∠3 and ∠4 17. ∠3 and ∠1 not adj. only adj. Extra Practice J A Tell whether the angles are only adjacent, adjacent and form a linear pair, or not adjacent. Independent Practice For See Exercises Example / - , 6 25. Probability The angle measures 30°, 60°, 120°, and 150° are written on slips of paper. You choose two slips of paper at random. What is the probability that the angle measures are supplementary? 1 _ 3 Multi-Step ∠ABD and ∠BDE are supplementary. Find the measures of both angles. 26. m∠ABD = 5x°, m∠BDE = (17x - 18)° 45°; 135° 27. m∠ABD = (3x + 12)°, m∠BDE = (7x - 32)° 72°; 108° 28. m∠ABD = (12x - 12)°, m∠BDE = (3x + 48)° 103.2°; 76.8° 32. The measure of an acute ∠ is less than 90°. Therefore the measure of its supp. must be between 90° and 180°, which means the supp. is an obtuse ∠. Multi-Step ∠ABD and ∠BDC are complementary. Find the measures of both angles. 29. m∠ABD = (5y + 1)°, m∠BDC = (3y - 7)° 61°; 29° 30. m∠ABD = (4y + 5)°, m∠BDC = (4y + 8)° 43.5°; 46.5° 31. m∠ABD = (y - 30)°, m∠BDC = 2y° 10°; 80° 32. Critical Thinking Explain why an angle that is supplementary to an acute angle must be an obtuse angle. 33. This problem will prepare you for the Multi-Step Test Prep on page 34. H is in the interior of ∠JAK. m∠JAH = (3x - 8)°, and m∠KAH = (x + 2)°. Draw a picture of each relationship. Then find the measure of each angle. a. ∠JAH and ∠KAH are complementary angles. m∠JAH = 64°; m∠KAH = 26° b. ∠JAH and ∠KAH form a linear pair. m∠JAH = 131.5°; m∠KAH = 48.5° c. ∠JAH and ∠KAH are congruent angles. m∠JAH = m∠KAH = 7° 1-4 PRACTICE A 1-4 PRACTICE C ,%33/. £{ *À>VÌViÊ 0AIRSOF!NGLES 1-4 PRACTICE B 32 012AND312FORMALINEARPAIR&INDTHESUMOFTHEIRMEASURES ,%33/. 5SETHEFIGURESFOR%XERCISESn SUPPLEMENTOF: COMPLEMENTOF9X X £{ 9 1-4 READING STRATEGIES £{ ,iÌi>V 0AIRSOF!NGLES 1-4 RETEACH !NGLE0AIRS !DJACENT!NGLES ADJACENTTWOANGLESINTHESAME PLANEWITHACOMMONVERTEXAND ACOMMONSIDEBUTNOCOMMON INTERIORPOINTS !NANGLEMEASURESDEGREESLESSTHANTHREETIMESITSSUPPLEMENT&INDTHE LINEARADJACENT ANGLESWHOSE NONCOMMONSIDES AREOPPOSITERAYS ,INEAR0AIRS HAVETHESAMEVERTEXAND SHAREACOMMONSIDE ANDAREADJACENT 6ERTICAL!NGLES NONADJACENTANGLESFORMED BYTWOINTERSECTINGLINES ADJACENTANGLESWHOSE NONCOMMONSIDESARE OPPOSITERAYS ge07se_c01_0028_0033.indd 32 !NANGLEISITSOWNCOMPLEMENT&INDTHEMEASUREOFASUPPLEMENTTOTHISANGLE COMPLEMENTARYTWO ANGLESWHOSEMEASURES HAVEASUMOF !NGLEPAIRS CANBE $%&AND&%'ARECOMPLEMENTARYM$%&X ANDM&%'X &INDTHEMEASURESOFBOTHANGLESM$%& M&%' &INDTHEMEASURESOFBOTHANGLESM$%& M&%' ??k +,-AND-,.ARECOMPLEMENTARY,- BISECTS+,.&INDTHEMEASURESOF+,- AND-,. ADJACENTANDFORM ALINEARPAIR ONLYADJACENT ANDONLYADJACENT ANDNOTADJACENT VERTICAL ANDADJACENTANDFORMALINEARPAIR NOTADJACENT COMPLEMENTARY RIGHT !"#AND#"$FORMALINEARPAIRANDHAVE EQUALMEASURES4ELLIF!"#ISACUTERIGHT OROBTUSE 4ELLWHETHERTHEINDICATEDANGLESAREONLYADJACENTAREADJACENTAND FORMALINEARPAIRORARENOTADJACENT 0OSSIBLEANSWERAND ANDAREVERTICAL ANGLES 4ELLWHETHERANDINEACHFIGUREAREONLYADJACENTAREADJACENTANDFORMA LINEARPAIRORARENOTADJACENT )DENTIFYEACHPAIROFANGLESASCOMPLEMENTARYSUPPLEMENTARYLINEAR VERTICALORADJACENT5SETHEGRAPHICORGANIZERABOVETOHELPYOU +EEPINMINDTHATTHEREMAYBEMORETHANONEANSWERFOREACHEXERCISE 12/3/05 11:50:11 AM ANDAREADJACENT ANDFORMALINEARPAIR .AMEALINEARPAIROFANGLES 5SETHEFIGUREFOR%XERCISESn )NSEVERALNICKELSWEREMINTEDTOCOMMEMORATETHE,OUISIANA 0URCHASEAND,EWISAND#LARKSEXPEDITIONINTOTHE!MERICAN7EST/NE NICKELSHOWSAPIPEANDAHATCHETCROSSEDTOSYMBOLIZEPEACEBETWEEN THE!MERICANGOVERNMENTAND.ATIVE!MERICANTRIBES .AMEAPAIROFVERTICALANGLES VERTICALTWO NONADJACENT ANGLESFORMED BYTWOINTER SECTINGLINES SUPPLEMENTARYTWO ANGLESWHOSEMEASURES HAVEASUMOF $%&AND&%'ARESUPPLEMENTARYM$%&X ANDM&%'X 0OSSIBLEANSWERAND ,%33/. 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