3-8 Perpendicular Lines
Constructing Parallel andPerpendicular Lines3-83-81. PlanGO for HelpWhat You’ll LearnCheck Skills You’ll Need To construct parallel lines To construct perpendicularUse a straightedge to draw each figure. Then use a straightedge and compassto construct a figure congruent to it. 1–3. See back of book.1. a segment2. an obtuse angle3. an acute anglelines. . . And WhyLesson 1-7Use a straightedge to draw each figure. Then use a straightedge and compassto bisect it. 4–6. See back of book.4. a segment5. an acute angle6. an obtuse angleTo construct the shortestsegment from a point to aline, as in Example 4Objectives12Examples12341To construct parallel linesTo construct perpendicularlinesConstructing 6 mConstructing a SpecialQuadrilateralPerpendicular at a Pointon a LinePerpendicular From a Pointto a LineConstructing Parallel LinesMath BackgroundYou can use what you know about parallel lines, transversals, and correspondingangles to construct parallel lines.1EXAMPLEConstructing 6 mConstruct the line parallel to a given line and through a given point that is not onthe line.NGiven: line / and point N not on /Construct: line m through N with m 6 / Step 1Label two points H and J on O.*)NDraw HN . HJMore Math Background: p. 124DStep 2Real-WorldConnectionCareers Architects constructparallel and perpendicularlines when they build modelsof the buildings they design.Construct &1 with vertex at N so that&1 &NHJ and the two angles arecorresponding angles. Label the lineyou just constructed m.mN 1Lesson Planning andResources Hm6OQuick CheckThe method in this lesson forconstructing parallel lines is basedon the Converse of the Corresponding Angles Postulate inLesson 3-2. An alternative methodmight base a construction on theConverse of the Alternate InteriorAngles Theorem or on thetheorem In a plane, if two linesare perpendicular to the same line,then they are parallel to eachother. The method for constructingperpendicular lines is based onthe method for constructing theperpendicular bisector of asegment in Lesson 1-7.JSee p. 124E for a list of theresources that support this lesson.1 Critical Thinking Explain why lines O and m must be parallel.If corr. ' are O, the lines are zz by the Converse of Corr. ' Postulate.PowerPointFor many constructions, you will find it helpful to first visualize or sketch whatthe final figure should look like. This will often suggest the construction steps.In Example 2, a sketch is shown at the left of the example.Bell Ringer PracticeCheck Skills You’ll NeedFor intervention, direct students to:Lesson 3-8 Constructing Parallel and Perpendicular LinesSpecial NeedsBelow LevelL1For students who lack manual dexterity, seek out anduse alternative compasses.learning style: tactile181L2Ask volunteers to demonstrate and explain how tocopy an angle and a segment and how to constructthe perpendicular bisector of a line segment.learning style: visualConstructing Segmentsand AnglesLesson 1-7: Examples 1 and 2Extra Skills, Word Problems, ProofPractice, Ch. 1Constructing BisectorsLesson 1-7: Examples 3 and 5Extra Skills, Word Problems, ProofPractice, Ch. 1181
2. Teach2Guided InstructionProblem Solving HintDo the constructions in this lessonon the board.1EXAMPLEbBError PreventionAConstruct: quadrilateral ABYZ withAZ a, BY b, and AZ 6 BYStep 2ZEXAMPLE* )BaABa2.AcdAdditional ExamplesZStep 4Construct Y so that BY b. Then draw YZ.d1 Draw a vertical line and apoint not on the line.Demonstrate the construction ofExample 1. Check students’constructions.bBdQuadrilateral ABYZ has AZ a, BY b,and AZ 6 BY.cQuick CheckYaAZ2 Draw two segments. Label their lengths c and d. Construct a quadrilateral with onepair of parallel sides of lengths c and 2d. See above left.2Part1 2 Constructing Perpendicular LinesDBZStep 3* )Construct a ray parallel to AZ through B.PowerPoint2 Construct a quadrilateral withboth pairs of sides parallel.Z)Math TipThe constructed quadrilateral is atrapezoid.AaADraw a point B not on AZ . Then draw AB .In Step 2, make sure that studentsrealize that the angle constructedat vertex N must be a corresponding angle. If the congruent anglewere constructedon the opposite* )side of HN , the lines would notbe parallel.2bStep 1Construct AZ with length a.YaConstructing a Special QuadrilateralConstruct a quadrilateral with one pair of parallel sides of lengths a and b.aGiven: segments of lengths a and bFirst, draw a sketch ofthe figure.Visual LearnersEXAMPLEYou can construct perpendicular lines using a compass and a straightedge.3CEXAMPLEPerpendicular at a Point on a LineConstruct the perpendicular to a given line at a given point on the line.Given: point P on line O* )* ) Construct: CP with CP ' /PStep 1Put the compass point on point P. Draw arcsintersecting / in two points. Label the points A and B.For: Construction ActivityUse: Interactive Textbook, 3-8182Step 2Open the compass wider. With the compass tipon A, draw an arc above point P. APPBBChapter 3 Parallel and Perpendicular LinesAdvanced LearnersEnglish Language Learners ELLL4Have students use the methods in Example 3 toconstruct a square as simply as possible.182 Alearning style: tactileIn Example 3, the word perpendicular is a noun. Pointout that perpendicular may mean a perpendicular lineor segment. Similarly, parallel may mean a parallelline or segment.learning style: verbal
Guided InstructionStep 3CWithout changing the compass setting, place thecompass point on point B. Draw an arc that intersectsthe arc from Step 2. Label the point of intersection C.Auditory Learners APHave students work withpartners to do the constructionsin Examples 3 and 4, taking turnsexplaining the steps in eachconstruction.BStep 4* )Draw CP .C A* )CP ' OQuick Check* )* )* )P4B* )3 Use a straightedge to draw EF . Construct FG so that FG ' EF at point F.See margin.You will prove in Chapter 5 that the perpendicular segment is the shortest segmentfrom a point to a line. Here is its construction.4Perpendicular From a Point to a LineEXAMPLEConstruct the perpendicular to a given linethrough a given point not on the line.Given: line O and point R not on O* )* )Step 1Open your compass to a size greaterthan the distance from R to O. Withthe compass point on point R, drawan arc that intersects O at twopoints. Label the points E and F.Step 2Place the compass point on Eand make an arc.Step 3Keep the same compass setting. With thecompass tip on F, draw an arc that intersectsthe arc from Step 2. Label the point ofintersection G.You can draw large circlesusing a simple, large compass. EQuick CheckFR EResources Daily Notetaking Guide 3-8 L3 Daily Notetaking Guide 3-8—L1Adapted InstructionFRClosure EFExplain how to construct a lineparallel to a given line. Tell whichtheorem or postulate you use.Construct congruent corresponding angles; the Converse ofthe Corresponding Angles Post.FG*)*)* )* )*)4 Draw a line CX and a point Z not on CX . Construct ZB so that ZB ' CX .See back of book.Lesson 3-8 Constructing Parallel and Perpendicular LinesQuick Check4 Examine the construction.At* )what special point doesRG meet line ? the midpointof EFGE* )RG ' O3 Why does step 2 instruct youto open the compass wider? Withthe compass tip on A and thenon B, the same compass settingwould make arcs that intersectat point P on line . Step 4* )Draw RG .ConnectionThe compass setting in step 2does not have to be the sameas that in step 1. However, thecompass settings must be thesame in steps 2 and 3 and mustbe large enough that the arcsconstructed in these two stepsintersect.Additional ExamplesRRReal-WorldTeaching TipPowerPointR Construct: RG with RG ' OEXAMPLE1833.GEF183
EXERCISES3. PracticePractice and Problem SolvingAssignment GuideA1 A B 1-7, 14-16Practice by ExampleExample 12 A BFor more exercises, see Extra Skill, Word Problem, and Proof Practice.8-13, 17-26C Challenge27-36Test PrepMixed Review37-4041-47GO forHelp(page 181)In Exercises 1–4, draw a figure like the given one. Then construct the line through* )point J and parallel to AB. 1–4. See margin pp. 184–185.J1.2. AJABB3.Homework Quick Check4. ABJTo check students’ understandingof key skills and concepts, go overExercises 4, 10, 15, 21, 24.JABAlternative MethodExample 2Exercise 14 This exercise presents(page 182)another way to construct a lineparallel to a given line.For Exercises 5–7, draw two segments. Label their lengths a and b. Construct aquadrilateral with one pair of parallel sides as described. 5–7. See back of book.5. The sides have lengths a and b.Exercise 25 Have students discuss6. The sides have lengths 2a and b.the significance of the phrasemust be true.7. The sides have lengths a and 12 b.Example 3(pages 182, 183)In Exercises 8–9, *draw) a figure like the given one. Then construct the lineperpendicular to AB at point P. 8–9. See back of book.8.AP9.BABExample 4GPS Guided Problem Solving(page 183)L3L4EnrichmentIn Exercises 10–13, draw a figure like the* )given one. Then construct the linethrough point P and perpendicular to RS . 10–13. See back of book.10. P11.L2ReteachingSL1Adapted PracticePracticeNameClassL3DatePractice 3-7PSPRConstructing Parallel and Perpendicular LinesRConstruct a line perpendicular to line l through point Q.1.2.Q3.QQ12. 13.P RS Construct a line perpendicular to line l at point T.4.T 5.T 6. TRPSConstruct a line parallel to line l and through point K.7.8.K9.KBK Apply Your Skills Pearson Education, Inc. All rights reserved. For Exercises 10–15, use the segments at the right.10. Construct a quadrilateral with one pair of parallel sides of lengthsa and b.abc11. Construct a quadrilateral with one pair of parallel sides of lengthsb and c.12. Construct a square with side lengths of b.13. Construct a right triangle with leg lengths of a and c.14. Construct a right triangle with leg lengths of b and c.14. Draw an acute angle. Construct an angle congruent to your angle so that thetwo angles are alternate interior angles. (Hint: Think of the letter Z.)See margin, p.185.15. Writing Explain how to use the Converse of the Alternate Interior AnglesTheorem to construct a line parallel to a given line through a point not onthe line. (Hint: See Exercise 14.)Construct a O alt. int. l; then draw the n line.15. Construct an isosceles right triangle with leg lengths of a.184Chapter 3 Parallel and Perpendicular Lines4 zz AB1.2.A zz ABJJA1843. BB zz ABJ AB
4. Assess & Reteach16. Draw obtuse nABC with obtuse &B.a. Construct line / through point A so that / 6 BC. a–b. See backb. Construct line m through point C so that m 6 AB. of book.PowerPointLesson QuizFor Exercises 17–24, use the segments at the left. 17–24. See back of book.17. Draw a line m. Construct a segment of length b that is perpendicular to line m.aDraw a figure similar to the onegiven. Then completethe construction. Checkstudents’ constructions.18. Construct a rectangle with base b and height c.b19. Construct a square with sides of length a.c20. Construct a rectangle with one side length a and a diagonal length b.21. a. Construct a quadrilateral with a pair of parallel sides of length c.GPS b. Make a Conjecture What appears to be true about the other pair of sides inthe quadrilateral you constructed?c. Use a protractor, a ruler, or both to check the conjecture you made in part (b).Exercises 17–241. Construct a line* ) through D thatis parallel to XY .DX22. Construct a right triangle with legs of lengths a and b.23. Construct a right triangle with legs of lengths b and 12 b.D24. a. Construct a triangle with sides of lengths a, b, and c.b. Construct the midpoint of each side of the triangle.c. Form a new triangle by connecting the midpoints.d. Make a Conjecture How do the sides of the smaller triangle and the sides ofthe larger triangle appear to be related?e. Use a protractor, a ruler, or both to check the conjecture you made in part (d).* )25. Multiple Choice The diagram at the left shows the construction of line CPperpendicular to line / through point P. Which of the following must be true? DC A* ) * )* )CP 5 12ABB* )AC BCAC ' CBCB 6 ABPYXY2. Construct a quadrilateral withone pair of parallel sides oflengths p and q.pqQuadrilaterals may vary.Sample:p26. Paper Folding You can use paper folding to create a perpendicular to a givenline through a given point (Activity Lab, page 102). Fold the paper so that theline folds onto itself and the fold line contains the given point.q3. Construct the line perpendicular to line m at point Z.GOnlineZHomework HelpmVisit: PHSchool.comWeb Code: aue-0308ZCa. Draw a line m and a point W not on the line. Use paper folding to create theperpendicular to m through W. Label this fold line k. a–b. Check students’b. Next, fold the line perpendicular to k through W. Label this fold line p. work.c. What is true of p and m? Justify your answer. p zz m; in a plane, two lines #to a third are n.Draw a segment, DG. Construct a quadrilateral whose diagonals are bothcongruent to DG, bisect each other, and meet the additional condition given below.Describe the quadrilateral that you get. 27–28. See back of book.Challenge27. The diagonals are not perpendicular.lesson quiz, PHSchool.com, Web Code: aua-03084.ABm4. Construct the perpendicular toline n through point O.OnOn28. The diagonals are perpendicular.Lesson 3-8 Constructing Parallel and Perpendicular Lines18514.2J n AB1l1 O l2185
Alternative Assessment29–32.See margin.Have students work in pairs toconstruct a rectangle. Their workshould include a constructiondiagram and a written explanationof the steps they used.Construct a rectangle whose side lengths a and b meet the given condition.30. b 12 a29. b 2a31. b 13 a32. b 23 aConstruct a triangle whose side lengths a, b, and c meet the given conditions. Ifsuch a triangle is not possible, explain. 33, 35. See margin. 34. See back of book.33. a b c34. a b 2c35. a 2b 2c36. a b cNot possible; The shorter sideswould meet at a point on thelonger side, forming a segment.Test PrepTest PrepResourcesFor additional practice with avariety of test item formats: Standardized Test Prep, p. 193 Test-Taking Strategies, p. 188 Test-Taking Strategies withTransparenciesMultiple Choice37. In the construction shown at the right, the two arcswith centers A and B have the same radius. Whatmust be true of PQ? AA. PQ bisects AB.B. PQ 6 ABC. PQ ABD. PQ AQPAB38. Suppose you construct lines O, m, and n so thatO ' m and O 6 n. Which of the following is true? JF. m 6 nG. m 6 OH. n ' OJ. n ' m29.Short ResponseaaaQ39. Use a compass and straightedge to construct the following figure.a. Draw a line O and a point G not on O. Construct an arc centered at point Gto intersect O in two points. Label the points R and T. Draw GR and GT.b. Classify nRGT. Justify your response. a–b. See back of book.40. These pictures show steps for constructing a line parallel to a given line,but they are not necessarily in order.b30.I.GCbII.G CIII.G CIV.GC a. List the construction steps in the correct order. a–b. See margin.b. For any step that uses a compass, describe the location(s) of thecompass point.a31.Mixed ReviewbGO forHelpa32.Yes; the slopes are both 13,Are the lines parallel? Explain.and the y-intercepts are different.No; the slopes are different.41. y -4x - 342. y 12 x 143. x 3y -6y 4x 3y -2x - 14x 12y -6No; the slopes are different.Lesson 1-8Find the distance between the points to the nearest tenth.Lesson 3-744. W(8, -2) and Z(2, 6) 10Lesson 1-3a33.acba b c186A45. W(-4.5, 1.2) and Z(3.5, -2.8) 8.9b186Name the intersection of the planes.446. plane ABE and plane EBCD EB447. plane AFDE and plane FCD DFEBFDCChapter 3 Parallel and Perpendicular Lines35. Not possible; if a 2b 2c, then 2a 2b 2c or a b c. Theshorter sides wouldmeet at the midpoint ofthe longer side, forminga segment.40. [2] a. II, IV, III, Ib. (III): location ofcompass at pointsC and G; (I): sameas III and theintersection points4of CG with arcsdrawn in (III)[1] incorrect sequence ORincorrect location ofcompass point
Practice 3-7 Constructing Parallel and Perpendicular Lines Construct a line perpendicular to line l through point Q. 1. 2. 3. Construct a line perpendicular to line l at point T. 4. 5. 6. Construct a line parallel to line l and through point K. 7. 8. 9. For Ex
A) Rotating perpendicular lines result in parallel lines. B) The lines remain perpendicular only if rotated 180 . C) The lines remain perpendicular only if rotated 360 . D) Rotated perpendicular lines always remain perpendicular lines. Explanation: Rotated perpendicular lines always remain
Any two vertical lines are parallel. Postulate 18 Slopes of Perpendicular Lines In a coordinate plane, two nonvertical lines are perpendicular if and only if the product of their slopes is -1. The slopes of the two lines that are perpendicular are negative reciprocals of each other. Horizontal lines are perpendicular to vertical lines
All vertical lines are parallel. Notes: Perpendicular Lines and Slopes Two lines in the same plane that intersect to form right angles are perpendicular lines. Nonvertical lines are perpendicular if and only if their slopes are negative reciprocals. Vertical lines are perpendicular to horizontal lines. Notes: x y 2 4 2 2 2 y 2x 2 y .
3 Parallel and Perpendicular Lines 3.1 Pairs of Lines and Angles 3.2 Parallel Lines and Transversals 3.3 Proofs with Parallel Lines 3.4 Proofs with Perpendicular Lines 3.5 Equations of Parallel and Perpendicular Lines Tree House (p. 130) Kiteboarding (p. 143) Crosswalk (p. 154) Bike Path (p. 161) Gymnastics (p. 130) Bi
In a plane, if a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other line. If h k and j h, then j k. Proof Example 2, p. 150; Question 2, p. 150 Theorem 3.12 Lines Perpendicular to a Transversal Theorem In a plane, if two lines are perpendicular to the same line, then they are parallel to .
Look at each group of lines. Trace over any parallel lines with a coloured pencil: Lines, angles and shapes – parallel and perpendicular lines 1 2 3 Parallel lines are always the same distance away from each other at any point and can never meet. They can be any length and go in any direc on. ab c ab c Perpendicular lines meet at right angles.
Identify the lines as parallel, perpendicular, or neither. Unit 1Identifying Intersecting, Perpendicular, and Parallel Lines Intersecting lines are lines that cross each other at one point, called the point of intersection. X is the point of intersection of lines LM and NO. Perpendicular lines are two lines that form a right angle at the
5.1b – Parallel Lines and its Angle Relationships Target 5.2: Apply and prove statements using perpendicularity theorems 5.2a – Prove Theorems about Perpendicular Lines! 5.2b – Constructions: Perpendicular and Parallel Lines! Target 5.3 : Use parallel and perpendicular lines to wri
