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AnglesCheck your understanding of important skills.Name Use a Metric Ruler Use a centimeter ruler to measure.Find the length in centimeters.1.2.centimeterscentimeters Classify Angles Classify the angle. Write acute, right, or obtuse.3.4.5. Parts of a Whole Write a fraction for each shaded part.6.7.8.9.WITH Houghton Mifflin Harcourt Publishing CompanyTMThe Sunshine Skyway Bridge crosses over Tampa Bay,Florida. Bridges and other building structures canYmodel geometric figures. Be a Math Detective andinvestigate the bridge. Describe the geometric figuresyou see. Then classify the labeled angles and triangle.SGOOnlineKAssessment Options: Soar to Success MathChapter 11415

Vocabulary Builder Visualize ItReview WordsComplete the Bubble Map using review words.acutecircleobtuserayrightstraightvertexTypes ofAnglesPreview Wordsclockwisecounterclockwisedegree ( )protractor Understand VocabularyDraw a line to match each word with its definition.1.2.protractor In the same direction in which thehands of a clock movedegree ( )3.clockwise4.counterclockwisethe hands of a clock move A tool for measuring the size of anangle The unit used for measuring angles416GOOnline eStudent Edition Multimedia eGlossary Houghton Mifflin Harcourt Publishing Company In the opposite direction in which

Lesson 11.1NameAngles and Fractional Parts of a CircleEssential Question How can you relate angles and fractional partsof a circle?InvestigateInvestigateMaterialsfraction circlesn1A. Place a12 piece on the circle. Place the tip of the fractionpiece on the center of the circle. Trace the fraction piece.What figure is formed by the fraction piece?What parts of the fraction piece represent the rays ofthe angle?On what part of the circle is the vertex of the angle?11B. Shade the angle formed by the12 piece. Label it 12 .1C. Place the 12piece back on the shadedangle. Turn it counterclockwise.Counterclockwise is the directionopposite from the way the hands moveon a clock.Trace the fraction piece in its newposition. How many twelfths have you2traced in all? Label 12. Houghton Mifflin Harcourt Publishing CompanyD. Turn the fraction piececounterclockwise again and trace it.Label the total number of twelfths.Continue until you reach theshaded angle.What figure is formed by turning and tracing the fraction piece?1How many times did you need to turn the 12piece to make a circle?How many angles come together in the center of the circle?Chapter 11417

Draw Conclusions1. Compare the size of the angle formed by a 1 4 piece and the1size of the angle formed by apiece. Use a 1 piece and yourmodel on page 417 to help.2.124Synthesize Describe the relationship between the sizeof the fraction piece and the number of turns it takes to makea circle.Make ConnectionsYou can relate fractions and angles to the hands of a clock.Let the hands of the clock represent the rays of an angle. Each 5-minute mark1represents a12 turn clockwise.41811 12 121093487 6 515 minutes elapse.30 minutes elapse.The minute hand makes aThe minute hand makes aturn clockwise.turn clockwise.11 12 121093487 6 511 12 121093487 6 5MATHEMATICAL PRACTICESExplain how an angle1formed in a circle using a41fraction piece is like a 4 turn and15 minutes elapsing on a clock.45 minutes elapse.60 minutes elapse.The minute hand makes aThe minute hand makes aturn clockwise.turn clockwise. Houghton Mifflin Harcourt Publishing Company11 12 121093487 6 511 12 121093487 6 5

NameShare and ShowTell what fraction of the circle the shaded angle represents.1.2.4.3.5.6.Tell whether the angle on the circle shows a 1 4 , 12 , 3 4 , or1 full turn clockwise or counterclockwise.7.8.10.11. Houghton Mifflin Harcourt Publishing Company9.12.13.Susan watched the game from 1 P.M. to 1:30 P.M.Describe the turn the minute hand made.14.Compare the angles in Exercises 1 and 5.Does the position of the angle affect the size of the angle? Explain.11 12 121093487 6 511 12 121093487 6 5Chapter 11 Lesson 1419

Problem SolvingNMATHEMATICALPRACTICESModel Reason Make SenseSense or Nonsense?15. Whose statement makes sense? Whosestatement is nonsense? Explain your reasoning.The shaded anglerepresents 3 8 ofthe circle.The shaded anglerepresents 1 4 ofthe circle.Adam’s Statement For the statement that is nonsense, write a statement thatmakes sense. What is another way to describe the size of the angle? Explain.420FOR MORE PRACTICE:Standards Practice Book, pp. P207–P208 Houghton Mifflin Harcourt Publishing CompanyCarla’s Statement

Lesson 11.2NameDegreesEssential Question How are degrees related to fractional partsof a circle?CONNECT You can use what you know about anglesand fractional parts of a circle to understand anglemeasurement. Angles are measured in units calleddegrees. Think of a circle divided into 360 equal parts.1An angle that turns through360 of the circle measures1 degree.The symbol for degrees is .UNLOCK the ProblemThe angle between two spokes on the bicycle wheel10turns through360 of a circle. What is the measure ofthe angle between the spokes? What part of an angle does aspoke represent?Example 1 Use fractional parts to find theangle measure. Houghton Mifflin Harcourt Publishing Company1Each360 turn measures degree.1Ten360 turns measure degrees.So, the measure of the angle between the spokes is .MATHEMATICAL PRACTICESHow many degrees isthe measure of an angle that turnsthrough 1 whole circle? Explain. The Penny Farthing bicyclewas built in the 1800s.Chapter 11421

Example 2 Find the measure of a right angle.right angle symbolThink: Through what fraction of a circledoes a right angle turn?STEP 1 Write 1 4 as an equivalent fraction with 360 in thedenominator.154360Think: 4 3 9 5 36, so 4 3 5 360.90STEP 2 Write 360in degrees.To write an equivalentfraction, multiply thenumerator and denominatorby the same factor.1An angle that turns through360 of a circle measures .90An angle that turns through360 of a circle measures .So, a right angle measures .Try This! Find the measure of a straight angle.Through what fraction of a circle does a straight angle turn?Write 12 as an equivalent fraction with 360 in the denominator.152 360Think: 2 3 18 5 36, so 2 3 5 360.1. How can you describe the measure of an acute angle in degrees?2. How can you describe the measure of an obtuse angle in degrees?422 Houghton Mifflin Harcourt Publishing CompanySo, a straight angle measures .

NameShare and ShowN1. Find the measure of the angle.Through what fraction of a circle does the angle turn?153 36013Think: 3 3 12 5 36, so 3 3 5 360.So, the measure of the angle is .Tell the measure of the angle in degrees.2.3.45360112MATHEMATICAL PRACTICESOn Your OwnNIf an angle measures60º, through what fraction of acircle does it turn? Explain.Tell the measure of the angle in degrees.4.5.360360110Classify the angle. Write acute, obtuse, right, or straight.6.7.8. Houghton Mifflin Harcourt Publishing Company127 9.180 37 Classify the triangle. Write acute, obtuse, or right.10.11.45 12.40 45 20 120 40 70 70 Chapter 11 Lesson 2423

MATHEMATICALPRACTICESModel Reason Make SenseUNLOCK the Problem13. Ava started reading at 3:30 P.M. She stopped for a snackat 4:15 P.M. During this time, through what fraction of acircle did the minute hand turn? How many degrees didthe minute hand turn?11 12 121093487 6 511 12 121093487 6 5a. What are you asked to find?–b. What information can you use to find the fraction of a circlethrough which the minute hand turned?c. How can you use the fraction of a circle through which theminute hand turned to find how many degrees it turned?d. Show the steps to solve the problem.STEP 1?33543360e. Complete the sentences.From 3:30 P.M. to 4:15 P.M., the minutehand made a turn clockwise.14.33 90 53604 3 90Is this anglemeasure obtuse? Explain.The minute hand turned degrees.15. Test Prep How many degrees are in anangle that turns through 2 of a circle?4A90ºB 180ºC 270ºD 360º424FOR MORE PRACTICE:Standards Practice Book, pp. P209–P210 Houghton Mifflin Harcourt Publishing CompanySTEP 2

Lesson 11.3NameMeasure and Draw AnglesEssential Question How can you use a protractor to measure anddraw angles?DUNLOCK the ProblemEmma wants to make a clay sculpture of herdaughter as she appears in the photo from herdance recital. How can she measure /DCE, orthe angle formed by her daughter’s arms?CEA protractor is a tool for measuring the size ofan angle.Activity Measure /DCE using a protractor.Materials protractorSTEP 1 Place the center point of the protractor onvertex C of the igncenter pointand 004080100145001370601100125STEP 2 Align the 0º mark on the scale of theprotractor with ray 16016020MMCM15EAlign bottom ray and 0 .Read the angle measure on that scale.Extend the ray if you need to.014040Read 201017012060300180110700arms is .100801MMCM19015So, the angle formed by Emma’s daughter’s308010040Read the m/DCE as the“measure of angle DCE”.5070601100120The m/DCE 5 .D14 Houghton Mifflin Harcourt Publishing CompanySTEP 3 Find where ray CD intersects the same scale.EMATHEMATICAL PRACTICESCan you line upeither ray of the angle with theprotractor when measuring?Explain.Chapter 11425

Draw Angles You can also use a protractor to drawan angle of a given measure.Activity Draw /KLM with a measure of 82º.Materials protractorSTEP 1 Use the straight edge of the protractor to draw andlabel ray LM.STEP 2 Place the center point of the protractor on point L.Align ray LM with the 0º mark on the protractor.STEP 3 Using the same scale, mark a point at 82º. Label thepoint K.STEP 4 Use the straight edge of the protractor to draw ray LK.Share and ShowA1. Measure /ABC.Place the center of the protractor on point 016016020Read where intersects the same scale.015Align ray BC with .So, the m/ABC is .Use a protractor to find the angle measure.3.ONMm/ONM 5TRBe sure to use the correctscale on the protractor.Ask yourself: Is themeasure reasonable?Sm/TSR 5Use a protractor to draw the angle.4. 170º4265. 78ºMATHEMATICAL PRACTICESDescribe howdrawing and measuringangles are similar. Houghton Mifflin Harcourt Publishing Company2.

NameOn Your OwnNUse a protractor to find the angle measure.6.7.QZXRYSm/QRS 5m/XYZ 5Use a protractor to draw the angle.8. 115º9. 67º Houghton Mifflin Harcourt Publishing CompanyDraw an example of each. Label the angle with its measure.10. an acute angle11. an obtuse angle12. a straight angle13. a right angleDraw an angle with a measure of 0 .14.Describe your drawing.Chapter 11 Lesson 3427

MATHEMATICALPRACTICESModel Reason Make SenseProblem Solving15. Mrs. Murphy is building a wheelchair ramp outside herbusiness. The angle of the ramp should be 5 . Draw a picturein the space to the right to show a model of the ramp.16.What’s the Error? Tracy measured an angle as 50º thatwas actually 130º. Explain her error.17. Test Prep What is the measure of /QRS?A 35ºB 45ºC 135ºSD 155ºRQEarth’s AxisEarth revolves around the sun yearly. The NorthernHemisphere is the half of Earth that is north of the equator.The seasons of the year are due to the tilt of Earth’s axis.Northern Hemisphere18. In the Northern Hemisphere, Earth’s axis is tilted awayfrom the sun on the first day of winter, which is often onDecember 21. What is the measure of the marked angleon the first day of winter, the shortest day of the year?WintersunEarth19. Earth’s axis is not tilted away from or toward the sunon the first days of spring and fall, which are often onMarch 20 and September 22. What is the measure of themarked angle on the first day of spring or fall?Spring and FallsunEarth20. In the Northern Hemisphere, Earth’s axis is tilted towardthe sun on the first day of summer, which is often onJune 21. What is the measure of the marked angle onthe first day of summer, the longest day of the year?428FOR MORE PRACTICE:Standards Practice Book, pp. P211–P212SummersunEarth Houghton Mifflin Harcourt Publishing CompanyUse the diagrams and a protractor for 18–20.

NameMid-Chapter Checkpoint VocabularyVocabularyChoose the best term from the box.clockwise1. The unit used to measure an angle is calledcounterclockwisea . (p. 421)2.degree (º)is the opposite of theprotractordirection in which the hands of a clock move. (p. 417)3. Ais a tool for measuring the sizeof an angle. (p. 425) Concepts and SkillsTell whether the angle on the circle shows a 1 4 , 12 , 3 4 , or 1 fullturn clockwise or counterclockwise.4.5.6.7.Tell the measure of the angle in degrees. Houghton Mifflin Harcourt Publishing Company8.1003609.112Use a protractor to draw the angle.10. 75 11. 127 Chapter 11429

Fill in the bubble completely to show your answer.12. Phillip watched a beach volleyball game from 1:45 P.M. to2:00 P.M. How many degrees did the minute hand turn?11 12 121093487 6 511 12 121093487 6 5A90 B 180 C 270 D 360 13. Which piece of pie forms a 180 angle?ACBDhelp you.TB430CA acute; 58 C obtuse; 62 B acute; 62 D obtuse; 118 Houghton Mifflin Harcourt Publishing Company14. Which best describes the m/CBT? Use a protractor to

Lesson 11.4NameJoin and Separate AnglesEssential Question How can you determine the measure of an angleseparated into parts?InvestigateInvestigateMaterials n construction paper n scissors n protractorA. Use construction paper. Draw an angle that measuresexactly 70 . Label it /ABC.B. Cut out /ABC.C. Separate /ABC by cutting it into two parts.Begin cutting at the vertex and cut betweenthe rays.What figures did you form?D. Use a protractor to measure the two angles you formed.Record the measures.E. Find the sum of the angles you formed.1 5part1part5 wholeF. Join the two angles. Compare the m/ABC to the sumYou can think of /ABCas the whole and the twoangles you formed as theparts of the whole. Houghton Mifflin Harcourt Publishing Companyof the measures of its parts. Explain how they compare.Chapter 11431

Draw Conclusions1. What if you cut /ABC into two different angles? What canyou conclude about the sum of the measures of these twoangles? Explain.2.Seth cut /ABC into 3 parts. Draw a model that showstwo different ways he could have separated his angle.3. Generalize Write a sentence that compares the measure of anangle to the sum of its parts.Make ConnectionsNBMaterials n protractorDYou can write the measure of the angles shownin a circle as a sum.ASTEP 2 Label each angle with its measure.MATHEMATICAL PRACTICESSTEP 3 Write the sum of the angle measures as an equation.1 1 5part4321part1part5 wholeDescribe the anglesshown in the circle above usingthe words whole and part. Houghton Mifflin Harcourt Publishing CompanySTEP 1 Use a protractor to find the measure of each angle.C

NameShare and ShowAdd to find the measure of the angle. Write anequation to record your work.1.2.P27 L43 37 Q3.10 55 STJm/PQT 90 K78 HRm/RHS m/JKL Use a protractor to find the measure of each angle.Label each angle with its measure. Write the sum ofthe angle measures as an equation.5.4.YXJKLM Houghton Mifflin Harcourt Publishing Company6.ZVWLook back at Exercise 1. Supposeyou joined an angle measuring 10º to /PQT. Describe anddraw the new angle. Include all three parts in your drawingand description.Chapter 11 Lesson 4433

MATHEMATICALPRACTICESModel Reason Make SenseUNLOCK the Problem7. Stephanie, Kay, and Shane each ate an equal-sized pieceof a pizza. The measure of the angle of each piece was 45º.When the pieces were together, what is the measure ofthe angle they formed?A 90ºB 135ºC 180ºD 225ºa. What are you asked to find?b. What information do you need to use?c. Tell how you can use addition to solve the problem.d. Fill in the bubble for the correct answer choice above.DR75 85 Q9. Which equation can you use to find them/XZW?TLYX42 71 WZA20ºB 150ºC 160ºD 180º434FOR MORE PRACTICE:Standards Practice Book, pp. P213–P214A 71º 2 42º 5nB 71º 1 42º 5nC 71º 3 42º 5nD 180º 2 113º 5n Houghton Mifflin Harcourt Publishing CompanyWhat is the m/QRT?8.

PROBLEM SOLVINGNameProblem Solving Unknown Angle MeasuresLesson 11.5Essential Question How can you use the strategy draw a diagram to solveangle measurement problems?UNLOCK the ProblemAMr. Tran is cutting a piece of kitchen tile as shown at the right.He needs tiles with 45º angles to make a design. After thecut, what is the angle measure of the part left over? CanMr. Tran use both pieces in the design?Use the graphic organizer below to solve the problem.Dx45 BCRead the ProblemWhat do I need to find?What information do Ineed to use?How will I use theinformation?I need to findI can use the measures ofthe angles I know.I can draw a bar model anduse the information toSolve the ProblemI can draw a bar model to represent the problem.Then I can write an equation to solve the problem. Houghton Mifflin Harcourt Publishing Company45 x90 m/ABD 1 m/CBD 5 m/ABCMATHEMATICAL PRACTICESWhat otherequation can you write tosolve the problem? Explain.x1 5x5The m/ABD .Since both tiles measure , Mr. Tran can useboth pieces in the design.Chapter 11435

Try Another ProblemMarisol is building a frame for a sandbox, but theboards she has are too short. She must join two boardstogether to build a side as shown. At what angle didshe cut the first board?LHx 55 JKRead the ProblemWhat do I need to find?What information do Ineed to use?How will I use theinformation? Explain how you can check the answer to the problem.436 Houghton Mifflin Harcourt Publishing CompanySolve the Problem

NameShare and Show1. Laura cuts a square out of scrap paper as shown.QMWhat is the angle measure of the piece left over?First, draw a bar model to represent the problem.115 PxNNext, write the equation you need to solve.Last, find the angle measure of the piece left over.The m/MNQ 5 .So, the angle measure of the piece left over is .QWhat if Laura cut a smaller square as shown?Would the m/MNQ be different? Explain.115 P225 xStreetThe map shows Marco’s paper route. WhenMarco turns right onto Center Street from Main Street,what degree turn does he make? Hint: Draw a dashedline to extend Oak Street to form a 180º angle.ter4.N125 Oak StreetxMain Street Houghton Mifflin Harcourt Publishing Companystraight edge as shown. What is the measure of thepiece she trimmed off?xCity Avenue3. Jackie trimmed a piece of scrap metal to make aMCen2.West StreetChapter 11 Lesson 5437

MATHEMATICALPRACTICESOn Your Own5.Two angles form a straightangle. One angle measures 89º. What is themeasure of the other angle? Explain.6.Pose a Problem Look back at Problem5. Write a similar problem about two anglesthat form a right angle.Model Reason Make Sense7. Sam paid 20 for two t-shirts. The price ofeach t-shirt was a multiple of 5. What are thepossible prices of the t-shirts?8. Zayna has 3 boxes with 15 art books in eachbox. She has 2 bags with 11 math books ineach bag. If she gives 30 books away, howmany art and math books does she have now?What’s the Question? It measuresgreater than 0º and less than 90º.10. Test Prep What is the unknown anglemeasure?A 22ºB 68ºC 90ºD 158º43822 XFOR MORE PRACTICE:Standards Practice Book, pp. P215–P216 Houghton Mifflin Harcourt Publishing Company9.

NameChapter Review/Test VocabularyVocabularyChoose the best term from the box.clockwise1. The size of an angle can be measured using a tool calledcounterclockwisea . (p. 425)protractor2. is the direction in which the hands ofa clock move. (p. 418) Concepts and SkillsTell what fraction of the circle the shaded anglerepresents.3.4.5.Use a protractor to draw the angle. Houghton Mifflin Harcourt Publishing Company6. 68 7. 145 8. Use a protractor to find the measure of each angle.OLabel each angle with its measure. Write the sumof the angle measures as an equation.MNPGOAssessment OptionsOnline Chapter TestChapter 11439

Fill in the bubble completely to show your answer.9. Which describes the turn the angle on the circle shows?A 90º clockwiseB 90º counterclockwiseC 180º clockwiseD 180º counterclockwise10. Which best describes the m/RST? Use a protractor to help you.RSTA acute; 48ºB obtuse; 48ºC obtuse; 132ºD obtuse; 148º11. The pocket watch was invented in 1524. The time is 6 P.M. After440A 45ºC 180ºB 90ºD 360º Houghton Mifflin Harcourt Publishing Company1 hour, how many degrees does the minute hand turn?

NameFill in the bubble completely to show your answer.12. What is the unknown angle measure?65 xA 25ºB 115ºC 125ºD 180º13. Which equation can you use to find the m/WRT?W84 69 RTA 84º 1 69º 5 nn84º 3 69º 5 nB 84º 2 69º 5CD 180º 2 153º 5 n14. If an angle measures 100º, through what fraction of a circle Houghton Mifflin Harcourt Publishing Companydoes the angle turn?1A100B 14100C360D 12Chapter 11441

Constructed Response15. How many right angles are there in an angle that turns through360º? Explain how you know.16. Soccer practice began at 2:30 P.M. and stopped at 3:00 P.M.because of rain. During this time, through what fraction of acircle did the minute hand turn? How many degrees did theminute hand turn? Explain. Performance Task17. Charlotte divided a whole pizza into 4 pieces. One pieceformed a straight angle. One piece formed a right angle. Twopieces formed acute angles with the same degree measure.B Label each angle with its degree measure.C Label each angle as a fraction of a circle.D Write an equation that represents the degree measure of thewhole pizza as the sum of the measures of its parts.442 Houghton Mifflin Harcourt Publishing CompanyA Draw angles to represent the 4 pieces.

Think: Through what fraction of a circle does a right angle turn? _ An angle that turns through _1 360 of a circle measures _. An angle that turns through _90 360 of a circle measures _. So, a right angle measures _. Through what fraction of a circle does a straight angle turn? _ Write 1_ 2 as an

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