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Angles of Elevation andDepression8-58-51. PlanWhat You’ll LearnCheck Skills You’ll NeedLesson 6-1Objectives To use angles of elevationGO for HelpRefer to rectangle ABCD to completethe statements.A7 5 BExamplesand depression to solveproblems2. &5 j l111. &1 j l7. . . And Why10 313. &3 j l64. m&1 m&5 j 90 111D5. m&10 m&3 j 180 6. &10 j l8To use the angle of elevationto calculate the height of anatural wonder, as inExample 216 8C23To use angles of elevation anddepression to solve problemsIdentifying Angles ofElevation and DepressionReal-World ConnectionReal-World ConnectionNew Vocabulary angle of elevation angle of depressionMath Background1Indirect measurement has beenused since antiquity to measuredistances that could not bemeasured directly. For example,Eratosthenes measured theEarth’s circumference more than2000 years ago, assuming theEarth to be round althoughsubsequent scholars assumedit to be flat.Using Angles of Elevation and DepressionSuppose a person on the ground seesa hot-air balloon gondola at a 388 angleabove a horizontal line.This angle is the angle of elevation.Horizontal lineAt the same time, a person in the hot-airballoon sees the person on the ground ata 388 angle below a horizontal line.Angle ofdepression38ⴗMore Math Background: p. 414DThis angle is the angle of depression.Angle ofelevationExamine the diagram. The angle of elevationis congruent to the angle of depressionbecause they are alternate interior angles.Lesson Planning andResources38ⴗHorizontal lineSee p. 414E for a list of theresources that support this lesson.1EXAMPLEIdentifying Angles of Elevation and DepressionPowerPointDescribe each angle as it relatesto the situation shown.a. &1b. &4Quick CheckBell Ringer Practice&1 is the angle ofdepression from thepeak to the hiker.&4 is the angle ofelevation from thehut to the hiker.Check Skills You’ll Need1For intervention, direct students to:Finding Measures of Angles23Lesson 3-1: Examples 4 and 5Extra Skills, Word Problems, ProofPractice, Ch. 31 Describe each angle asit relates to the situationin Example 1.a. &2b. &3 l of depression from hiker to hutl of elevation from hiker to peakApplying the TriangleAngle-Sum Theorem4Lesson 8-5 Angles of Elevation and DepressionSpecial NeedsBelow LevelL1Use different colors to indicate angles of elevationand angles of depression. Then have students statethe angle of elevation or depression from what objectto what object.learning style: visualLesson 3-4: Example 1Extra Skills, Word Problems, ProofPractice, Ch. 3445L2Highlight the importance of parallel lines by havingstudents copy the diagrams in Examples 1 and 3 andmarking pairs of congruent angles in different colors.learning style: visual445

2. TeachSurveyors use two instruments, the transit and the theodolite,to measure angles of elevation and depression. On bothinstruments, the surveyor sets the horizon line perpendicularto the direction of gravity. Using gravity to find the horizonline ensures accurate measures even on sloping surfaces.Guided Instruction2EXAMPLECareers2Have students research the workdescription and tools of surveyors,including electronic distancemeasurement devices (EDMs).Additional Examples48 36 ft1 Describe &1 and &2 as they5 ftrelate to the situation shown.Pull ofgravityConnectionSurveying To find the height ofDelicate Arch in Arches NationalPark in Utah, a surveyor levels atheodolite with the bottom of thearch. From there, she measuresthe angle of elevation to the topof the arch. She then measures thedistance from where she stands toa point directly under the arch. Herresults are shown in the diagram.What is the height of the arch?x ftPowerPointReal-WorldEXAMPLEHorizonlinenot to scalextan 48 36x 36(tan 48 )1236Use the tangent ratio.Solve for x.3 9 . 9 8205148Use a calculator.So x 40. To find the height of the arch, add the height of the theodolite.Since 40 5 45, Delicate Arch is about 45 feet high.l1 is the angle of depression;l2 is the angle of elevation.2 A surveyor stands 200 ft froma building to measure its heightwith a 5-ft tall theodolite. Theangle of elevation to the top ofthe building is 35 . How tall isthe building? about 145 ftQuick Check34above ground begins a 2 descentto land at an airport. How manymiles from the airport is theairplane when it starts its descent?about 19 mi5AAEEDCBCBEDCBEDCBADCBA2CBA133 An airplane flying 3500 ft2 You sight a rock climber on a cliff at a 328 angle of elevation.The horizontal ground distance to the cliff is 1000 ft.Find the line-of-sight distance to the rock climber.Personabout 1179 ftDDEETest-Taking TipFor problems withangles of elevation ordepression, draw adetailed diagram tohelp you visualize thegiven information.Resources Daily Notetaking Guide 8-5 L3 Daily Notetaking Guide 8-5—L1Adapted InstructionReal-WorldEXAMPLE5.7 mi6.2 miThe airplane is 2714 - 1007, or 1707 ft abovethe level of the airport.sin 38 1707x17075280Closure3 Angle of descent2714 ftnot toscaleAltitude ofairport: 1007 ft9.8 mi3 1707 ftx3 Use the sine ratio.x 1707 sin 332 1000 ftConnectionMultiple Choice To approach runway 17 of thePonca City Municipal Airport in Oklahoma, thepilot must begin a 38 descent starting from analtitude of 2714 ft. The airport altitude is 1007 ft.How many miles from the runway is the airplaneat the start of this approach?3.6 miClimberSolve for x.32616.26.17731053Use a calculator.Divide by 5280 to convert feet to miles.The airplane is about 6.2 mi from the runway. The correct answer is C.Two buildings are 30 ft apart.The angle of elevation from thetop of one to the top of the otheris 19 . What is their difference inheight? about 10 ftQuick Check4463 An airplane pilot sights a life raft at a 268 angle of depression. The airplane’saltitude is 3 km. What is the airplane’s surface distance d from the raft?about 6.2 kmChapter 8 Right Triangles and TrigonometryAdvanced LearnersEnglish Language Learners ELLL4Challenge students to solve Example 3 using thecosine ratio.446learning style: verbalRelate the meaning of angle of depression todepressions in the terrain or the Great Depression.Relate the meaning of angle of elevation to anelevator or elevation.learning style: verbal

EXERCISESFor more exercises, see Extra Skill, Word Problem, and Proof Practice.3. PracticePractice and Problem SolvingAssignment GuideAPractice by ExampleExample 1GO forHelpDescribe each angle as it relates to the situation in the diagram. 1–8. See margin.1. &12. &23. &34. &45. &56. &67. &78. &8(page 445)45110. 502.4 m9. 34.2 ft100 ftx203 m22 x11. Meteorology A meteorologist measures the angle of elevation of a weatherballoon as 418. A radio signal from the balloon indicates that it is 1503 m fromhis location. To the nearest meter, how high above the ground is the balloon?about 986 mFind the value of x. Round the lengths to the nearest tenth of a unit.12.580 yd27 31-3435-40Error Prevention!KelleyFind the value of x. Round the lengths to the nearest tenth of a unit.20 (page 446)Test PrepMixed ReviewTo check students’ understandingof key skills and concepts, go overExercises 10, 14, 19, 24, 26.6JimExample 329-30Homework Quick Check82(page 446)C Challenge73Example 21 A B 1-28263.3 yd13.18 0.6 kmxxExercise 14 Some students maythink the angle of depression isthe angle between the verticalsegment to the ground and theship. Ask each student to drawa diagram that represents thesituation in the exercise and thencompare diagrams with a partner.Emphasize that one side of anangle of depression or of an angleof elevation must be horizontal.2 km14. Indirect Measurement Miguel looks out from the crown of the Statue ofLiberty approximately 250 ft above ground. He sights a ship coming intoNew York harbor and measures the angle of depression as 188. Find thedistance from the base of the statue to the ship to the nearest foot. 769 ftApply Your Skills15. Flagpole The world’s tallest unsupported flagpole is a 282-ft-tall steel pole inSurrey, British Columbia. The shortest shadow cast by the pole during the yearis 137 ft long. To the nearest degree, what is the angle of elevation of the sunwhen the shortest shadow is cast? 64 L2ReteachingL1Adapted PracticePracticeName16. Engineering The Americans with Disabilities Act states that wheelchair ramps1can have a slope no greater than 12. Find the angle of elevation of a ramp withthis slope. Round your answer to the nearest tenth. 4.8 GOnlineHomework HelpVisit: PHSchool.comWeb Code: aue-080517. Construction Two office buildings are51 m apart. The height of the tallerbuilding is 207 m. The angle ofdepression from the top of the tallerbuilding to the top of the shorterbuilding is 158. Find the height of theshorter building to the nearest meter.about 194 m15 L3L4EnrichmentClassL3DatePractice 8-5Proportions in TrianglesJUse the figure at the right to complete each proportion.?1. AD EHDGCF FI2. BE?3. JA AB?JCJF ?4. FEDE?5. GHHI ?6. AD BHAGA B CDEGFHIAlgebra Find the values of the variables.7.51 xnot to scale12.x5–310x28207 m21x5x Pearson Education, Inc. All rights reserved.BGPS Guided Problem Solvingxy202122y125Lesson 8-5 Angles of Elevation and Depression447Algebra Solve for x.16.x17.xⴙ118.xxⴙ469xⴚ1xⴙ21. l of elevation from subto boat2. l of depression fromboat to sub3. l of elevation from boatto lighthouse4. l of depression fromlighthouse to boat5. l of elevation from Jimto waterfallxxⴙ52x ⴚ 8xⴙ86. l of elevation fromKelley to waterfall7. l of depression fromwaterfall to Jim8. l of depression fromwaterfall to Kelley447