3 Parallel And Perpendicular Lines - Mrfennmathclass

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33.13.23.33.43.5Parallel andPerpendicular LinesPairs of Lines and AnglesParallel Lines and TransversalsProofs with Parallel LinesProofs with Perpendicular LinesEquations of Parallel and Perpendicular LinesBike Path (p. 161)Crosswalk (p.(p 154)Kiteboarding (p. 143)SEE the Big IdeaGymnastics (p. 130)TreeTree HHouseouse (p.(p. 130)130)hs geo pe 03co.indd 1221/19/15 9:19 AM

Maintaining Mathematical ProficiencyFinding the Slope of a LineExample 1 Find the slope of the line shown.Let ( x1, y1 ) ( 2, 2) and ( x2, y2 ) (1, 0).4y2 y1slope —x2 x12Write formula for slope.0 ( 2) —1 ( 2)Substitute.2 —3Simplify. 4 2y(1, 0)224x3( 2, 2)Find the slope of the line.y1.2.3( 1, 2) 34( 2, 2)y3.4y221 1x1(3, 1) 4 224x 4 2 3 22( 3, 2)( 3, 1)4x(1, 2) 4Writing Equations of LinesExample 2 Write an equation of the line that passes through the point ( 4, 5)and has a slope of —34.y mx b5 3—4 ( 4) bWrite the slope-intercept form.Substitute —34 for m, 4 for x, and 5 for y.5 3 bSimplify.8 bSolve for b.So, an equation is y —34 x 8.Write an equation of the line that passes through the given point andhas the given slope.4. (6, 1); m 317. (2, 4); m —25. ( 3, 8); m 26. ( 1, 5); m 418. ( 8, 5); m —429. (0, 9); m —310. ABSTRACT REASONING Why does a horizontal line have a slope of 0, but a vertical line hasan undefined slope?Dynamic Solutions available at BigIdeasMath.comhs geo pe 03co.indd 1231231/19/15 9:19 AM

MathematicalPracticesMathematically proficient students use technological tools to explore concepts.Characteristics of Lines in a Coordinate PlaneCore ConceptLines in a Coordinate Plane1.In a coordinate plane, two lines are parallel if and only if they are both vertical linesor they both have the same slope.2.In a coordinate plane, two lines are perpendicular if and only if one is vertical and theother is horizontal or the slopes of the lines are negative reciprocals of each other.3.In a coordinate plane, two lines are coincident if and only if their equationsare equivalent.Classifying Pairs of LinesHere are some examples of pairs of lines in a coordinate plane.a.2x y 2x y 4These lines are not parallelor perpendicular. Theyintersect at (2, 2).b.2x y 2 These lines are coincident4x 2y 4 because their equationsare equivalent.44 6 666 4c.2x y 22x y 4 4These lines are parallel.Each line has a slopeof m 2.d. 2x y 2x 2y 44These lines are perpendicular.They have slopes of m1 2and m2 —12 .4 6 666 4 4Monitoring ProgressUse a graphing calculator to graph the pair of lines. Use a square viewing window. Classify the linesas parallel, perpendicular, coincident, or nonperpendicular intersecting lines. Justify your answer.1. x 2y 22x y 4124Chapter 3hs geo pe 03co.indd 1242. x 2y 23. x 2y 22x 4y 4x 2y 24. x 2y 2x y 4Parallel and Perpendicular Lines1/19/15 9:19 AM

3.1Pairs of Lines and AnglesEssential QuestionWhat does it mean when two lines are parallel,intersecting, coincident, or skew?Points of IntersectionWork with a partner. Write the number of points of intersection of each pair ofcoplanar lines.a. parallel linesb. intersecting linesc. coincident linesClassifying Pairs of LinesWork with a partner. The figure shows aright rectangular prism. All its angles areright angles. Classify each of the following pairsof lines as parallel, intersecting, coincident,or skew. Justify your answers. (Two lines areskew lines when they do not intersect andare not coplanar.)Pair of LinesBCDAFEClassificationGIHReasona. ⃖ ⃗AB and ⃖ ⃗BC⃖ ⃗ and BC⃖ ⃗b. ADc. ⃖ ⃗EI and ⃖ ⃗IHd. ⃖ ⃗BF and ⃖ ⃗EHCONSTRUCTINGVIABLE ARGUMENTSTo be proficient in math,you need to understandand use stated assumptions,definitions, and previouslyestablished results.e. ⃖ ⃗EF and ⃖ ⃗CGf. ⃖ ⃗AB and ⃖ ⃗GHIdentifying Pairs of AnglesWork with a partner. In the figure, two parallel linesare intersected by a third line called a transversal.a. Identify all the pairs of vertical angles. Explainyour reasoning.b. Identify all the linear pairs of angles. Explainyour reasoning.5 61 28 74 3Communicate Your Answer4. What does it mean when two lines are parallel, intersecting, coincident, or skew?5. In Exploration 2, find three more pairs of lines that are different from thosegiven. Classify the pairs of lines as parallel, intersecting, coincident, or skew.Justify your answers.Section 3.1hs geo pe 0301.indd 125Pairs of Lines and Angles1251/19/15 9:21 AM

3.1LessonWhat You Will LearnIdentify lines and planes.Identify parallel and perpendicular lines.Core VocabulVocabularylarryparallel lines, p. 126skew lines, p. 126parallel planes, p. 126transversal, p. 128corresponding angles,p. 128alternate interior angles,p. 128alternate exterior angles,p. 128consecutive interior angles,p. 128Identify pairs of angles formed by transversals.Identifying Lines and PlanesCore ConceptParallel Lines, Skew Lines, and Parallel PlanesTwo lines that do not intersect are either parallel lines or skew lines. Two linesare parallel lines when they do not intersect and are coplanar. Two lines are skewlines when they do not intersect and are not coplanar. Also, two planes that do notintersect are parallel planes.Lines m and n are parallel lines (m n).kmPreviousperpendicular linesTnLines m and k are skew lines.Planes T and U are parallel planes (T U ).Lines k and n are intersecting lines, and thereis a plane (not shown) containing them.USmall directed arrows, as shown in red on lines m and n above, are used to showthat lines are parallel. The symbol means “is parallel to,” as in m n.Segments and rays are parallel when they lie in parallel lines. A line is parallelto a plane when the line is in a plane parallel to the given plane. In the diagramabove, line n is parallel to plane U.Identifying Lines and PlanesREMEMBERRecall that if two linesintersect to form a rightangle, then they areperpendicular lines.Think of each segment in the figure as part of a line.Which line(s) or plane(s) appear to fit the description?a. line(s) parallel to ⃖ ⃗CD and containing point Ab. line(s) skew to ⃖ ⃗CD and containing point Ac. line(s) perpendicular to ⃖ ⃗CD and containing point ADCBAFEGHd. plane(s) parallel to plane EFG and containing point ASOLUTIONa. ⃖ ⃗AB, ⃖ ⃗HG, and ⃖ ⃗EF all appear parallel to ⃖ ⃗CD, but only ⃖ ⃗AB contains point A.⃖ ⃗ and AH⃖ ⃗ appear skew to ⃖ ⃗b. Both AGCD and contain point A.⃖ ⃗ contains point A.c. ⃖ ⃗BC, ⃖ ⃗AD, ⃖ ⃗DE, and ⃖ ⃗FC all appear perpendicular to ⃖ ⃗CD, but only ADd. Plane ABC appears parallel to plane EFG and contains point A.Monitoring ProgressHelp in English and Spanish at BigIdeasMath.com1. Look at the diagram in Example 1. Name the line(s) through point F that appearskew to ⃖ ⃗EH.126Chapter 3hs geo pe 0301.indd 126Parallel and Perpendicular Lines1/19/15 9:21 AM

Identifying Parallel and Perpendicular LinesTwo distinct lines in the same plane either areparallel, like lineℓ and line n, or intersect in apoint, like line j and line n.kjPnThrough a point not on a line, there are infinitelymany lines. Exactly one of these lines is parallelto the given line, and exactly one of them isperpendicular to the given line. For example, line kis the line through point P perpendicular to lineℓ,and line n is the line through point P parallel to lineℓ.PostulatesPostulate 3.1 Parallel PostulatePIf there is a line and a point not on the line, thenthere is exactly one line through the point parallelto the given line.There is exactly one line through P parallel toℓ.Postulate 3.2 Perpendicular PostulateIf there is a line and a point not on the line,then there is exactly one line through the pointperpendicular to the given line.PThere is exactly one line through Pperpendicular toℓ.Identifying Parallel and Perpendicular LinesnePayThe given line markings show how theroads in a town are related to one another.AveSverOliBttreea. Name a pair of parallel lines.b. Name a pair of perpendicular lines.265C384ay TraSeawRlckWa9t hMAneAveverOli⃖ ⃗ ⃖ ⃗a. MDFEee tStr⃖ ⃗ ⃖ ⃗b. MDBFeatfWh265⃖ ⃗⃖ ⃗FE is not parallel to ⃖ ⃗AC, because MDFE, and by the Parallelis parallel to ⃖ ⃗384384429EStieldFynePac.AvedPayDilSOLUTIONNash Rdc. Is ⃖ ⃗FE ⃖ ⃗AC? Explain.429AvePostulate, there is exactly one lineparallel to ⃖ ⃗FE through M.Monitoring ProgressHelp in English and Spanish at BigIdeasMath.com2. In Example 2, can you use the Perpendicular Postulate to show that ⃖ ⃗AC is notperpendicular to ⃖ ⃗BF? Explain why or why not.Section 3.1hs geo pe 0301.indd 127Pairs of Lines and Angles1271/19/15 9:21 AM

Identifying Pairs of AnglesA transversal is a line that intersects two or more coplanar lines at different points.Core ConceptAngles Formed by Transversalst2t465Two angles are correspondingangles when they have correspondingpositions. For example, 2 and 6are above the lines and to the right ofthe transversal t.Two angles are alternate interiorangles when they lie between thetwo lines and on opposite sides ofthe transversal t.t1t358Two angles are alternate exteriorangles when they lie outside thetwo lines and on opposite sides ofthe transversal t.Two angles are consecutive interiorangles when they lie between thetwo lines and on the same side ofthe transversal t.Identifying Pairs of AnglesIdentify all pairs of angles of the given type.a.b.c.d.correspondingalternate interioralternate exteriorconsecutive interior5 67 81 23 4SOLUTIONa. l and 5 2 and 6 3 and 7 4 and 8b. 2 and 7 4 and 5Monitoring Progressc. l and 8 3 and 6d. 2 and 5 4 and 7Help in English and Spanish at BigIdeasMath.comClassify the pair of numbered angles.3.154.5.25 47128Chapter 3hs geo pe 0301.indd 128Parallel and Perpendicular Lines1/19/15 9:21 AM

3.1ExercisesDynamic Solutions available at BigIdeasMath.comVocabulary and Core Concept Check1. COMPLETE THE SENTENCE Two lines that do not intersect and are also not parallelare lines.2. WHICH ONE DOESN’T BELONG? Which angle pair does not belong with the other three?Explain your reasoning. 2 and 3 4 and 5 1 and 8 2 and 71 23 45 67 8Monitoring Progress and Modeling with MathematicsIn Exercises 3 –6, think of each segment in the diagramas part of a line. All the angles are right angles. Whichline(s) or plane(s) contain point B and appear to fit thedescription? (See Example 1.)CBDA⃖ ⃗? Explain.9. Is ⃖ ⃗PN KM⃖ ⃗? Explain.10. Is ⃖ ⃗PR NPIn Exercises 11–14, identify all pairs of angles of thegiven type. (See Example 3.)1 23 4GF5 67 8HE11. corresponding⃖ ⃗3. line(s) parallel to CD12. alternate interior⃖ ⃗4. line(s) perpendicular to CD13. alternate exterior5. line(s) skew to ⃖ ⃗CD14. consecutive interior6. plane(s) parallel to plane CDHUSING STRUCTURE In Exercises 15–18, classify theIn Exercises 7–10, use the diagram. (See Example 2.)angle pair as corresponding, alternate interior, alternateexterior, or consecutive interior angles.NMLKQSP9 1011 121 23 45 67 8R13 1415 167. Name a pair of parallel lines.15. 5 and 116. 11 and 138. Name a pair of perpendicular lines.17. 6 and 1318. 2 and 11Section 3.1hs geo pe 0301.indd 129Pairs of Lines and Angles1291/19/15 9:21 AM

ERROR ANALYSIS In Exercises 19 and 20, describeand correct the error in the conditional statementabout lines. 19. 20.24. HOW DO YOU SEE IT? Think of each segment in thefigure as part of a line.Ka. Which lines are⃖ ⃗?parallel to NQIf two lines do not intersect, thenthey are parallel.Nb. Which lines⃖ ⃗?intersect NQRPd. Should you have named all the lines on the cube⃖ ⃗? Explain.in parts (a)–(c) except NQ21. MODELING WITH MATHEMATICS Use the photo toIn Exercises 25–28, copy and complete the statement.List all possible correct answers.decide whether the statement is true or false. Explainyour reasoning.GEDDCSQc. Which lines areskew to ⃖ ⃗NQ?If there is a line and a point not onthe line, then there is exactly one linethrough the point that intersectsthe given line.LMABAFJCHB25. BCG and are corresponding angles.26. BCG and are consecutive interior angles.27. FCJ and are alternate interior angles.a. The plane containing the floor of the tree house isparallel to the ground.28. FCA and are alternate exterior angles.b. The lines containing the railings of the staircase,such as ⃖ ⃗AB, are skew to all lines in the planecontaining the ground.29. MAKING AN ARGUMENT Your friend claims theuneven parallel bars in gymnastics are not reallyparallel. She says one is higher than the other, so theycannot be in the same plane. Is she correct? Explain.c. All the lines containing the balusters, such as⃖ ⃗, are perpendicular to the plane containing theCDfloor of the tree house.22. THOUGHT PROVOKING If two lines are intersected bya third line, is the third line necessarily a transversal?Justify your answer with a diagram.23. MATHEMATICAL CONNECTIONS Two lines are cut bya transversal. Is it possible for all eight angles formedto have the same measure? Explain your reasoning.Maintaining Mathematical ProficiencyUse the diagram to find the measures ofall the angles. (Section 2.6)30. m 1 76 21431. m 2 159 130Chapter 3hs geo pe 0301.indd 130Reviewing what you learned in previous grades and lessons3Parallel and Perpendicular Lines1/19/15 9:21 AM

3.2Parallel Lines and TransversalsEssential QuestionWhen two parallel lines are cut by a transversal,which of the resulting pairs of angles are congruent?Exploring Parallel LinesWork with a partner.Use dynamic geometry softwareto draw two parallel lines. Drawa third line that intersects bothparallel lines. Find the measuresof the eight angles that areformed. What can you conclude?6D54B3E12F68 7524 31A0 3ATTENDING TOPRECISIONTo be proficient in math,you need to communicateprecisely with others. 2 101234C56Writing ConjecturesWork with a partner. Use the results of Exploration 1 to write conjectures aboutthe following pairs of angles formed by two parallel lines and a transversal.a. corresponding angles1 24 3b. alternate interior angles5 68 71 24 3c. alternate exterior angles1 24 35 68 7d. consecutive interior angles5 68 71 24 35 68 7Communicate Your Answer3. When two parallel lines are cut by a transversal, which of the resulting pairs ofangles are congruent?4. In Exploration 2, m 1 80 . Find the other angle measures.Section 3.2hs geo pe 0302.indd 131Parallel Lines and Transversals1311/19/15 9:23 AM

What You Will Learn3.2 LessonUse properties of parallel lines.Prove theorems about parallel lines.Core VocabulVocabularylarrySolve real-life problems.Previouscorresponding anglesparallel linessupplementary anglesvertical anglesUsing Properties of Parallel LinesTheoremsTheorem 3.1 Corresponding Angles TheoremIf two parallel lines are cut by a transversal, then the pairs of correspondingangles are congruent.t1 23 4Examples In the diagram at the left, 2 6 and 3 7.pProof Ex. 36, p. 180Theorem 3.2 Alternate Interior Angles Theorem5 67 8If two parallel lines are cut by a transversal, then the pairs of alternate interiorangles are congruent.qExamples In the diagram at the left, 3 6 and 4 5.Proof Example 4, p. 134Theorem 3.3 Alternate Exterior Angles TheoremIf two parallel lines are cut by a transversal, then the pairs of alternate exteriorangles are congruent.Examples In the diagram at the left, 1 8 and 2 7.Proof Ex. 15, p. 136Theorem 3.4 Consecutive Interior Angles TheoremIf two parallel lines are cut by a transversal, then the pairs of consecutive interiorangles are supplementary.Examples In the diagram at the left, 3 and 5 are supplementary, andANOTHER WAY 4 and 6 are supplementary.There are many waysto solve Example 1.Another way is to use theCorresponding AnglesTheorem to find m 5and then use the VerticalAngles CongruenceTheorem (Theorem 2.6)to find m 4 and m 8.Proof Ex. 16, p. 136Identifying AnglesThe measures of three of the numbered angles are120 . Identify the angles. Explain your reasoning.SOLUTION120º 23 45 67 8By the Alternate Exterior Angles Theorem, m 8 120 . 5 and 8 are vertical angles. Using the Vertical Angles Congruence Theorem(Theorem 2.6), m 5 120 . 5 and 4 are alternate interior angles. By the Alternate Interior Angles Theorem, 4 120 .So, the three angles that each have a measure of 120 are 4, 5, and 8.132Chapter 3hs geo pe 0302.indd 132Parallel and Perpendicular Lines1/19/15 9:23 AM

Using Properties of Parallel LinesFind the value of x.115 4(x 5) abSOLUTIONBy the Vertical Angles Congruence Theorem (Theorem 2.6), m 4 115 . Lines a andb are parallel, so you can use the theorems about parallel lines.Check115 (x 5) 180 ?115 (60 5) 180180 180m 4 (x 5) 180 Consecutive Interior Angles Theorem115 (x 5) 180 Substitute 115 for m 4.x 120 180 Combine like terms.x 60Subtract 120 from each side.So, the value of x is 60.Using Properties of Parallel LinesFind the value of x.1136 c(7x 9) dSOLUTIONBy the Linear Pair Postulate (Postulate 2.8), m 1 180 136 44 . Lines c and dare parallel, so you can use the theorems about parallel lines.m 1 (7x 9) Check44 (7x 9) ?44 7(5) 944 44 Alternate Exterior Angles Theorem44 (7x 9) Substitute 44 for m 1.35 7xSubtract 9 from each side.5 xDivide each side by 7.So, the value of x is 5.Monitoring ProgressHelp in English and Spanish at BigIdeasMath.comUse the diagram.1. Given m 1 105 , find m 4, m 5, andm 8. Tell which theorem you use in each case.2. Given m 3 68 and m 8 (2x 4) ,1 23 45 67 8what is the value of x? Show your steps.Section 3.2hs geo pe 0302.indd 133Parallel Lines and Transversals1331/19/15 9:23 AM

Proving Theorems about Parallel LinesProving the Alternate Interior Angles TheoremProve that if two parallel lines are cut by a transversal, then the pairs of alternateinterior angles are congruent.SOLUTIONSTUDY TIPBefore you write a proof,identify the Given andProve statements for thesituation described or forany diagram you draw.tDraw a diagram. Label a pair of alternateinterior angles as 1 and 2. You are looking foran angle that is related to both 1 and 2. Noticethat one angle is a vertical angle with 2 and acorresponding angle with 1. Label it 3.p12q3Given p qProve 1 2STATEMENTSREASONS1. p q1. Given2. 1 32. Corresponding Angles Theorem3. 3 23. Vertical Angles Congruence Theorem (Theorem 2.6)4. 1 24. Transitive Property of Congruence (Theorem 2.2)Monitoring ProgressHelp in English and Spanish at BigIdeasMath.com3. In the proof in Example 4, if you use the third statement before the secondstatement, could you still prove the theorem? Explain.Solving Real-Life ProblemsSolving a Real-life ProblemWhen sunlight enters a drop of rain,different colors of light leave the dropat different angles. This process iswhat makes a rainbow. For violet light,m 2 40 . What is m 1? How doyou know?21SOLUTIONBecause the Sun’s rays are parallel, 1 and 2 are alternate interior angles.By the Alternate Interior Angles Theorem, 1 2.So, by the definition of congruent angles, m 1 m 2 40 .Monitoring ProgressHelp in English and Spanish at BigIdeasMath.com4. WHAT IF? In Example 5, yellow light leaves a drop at an angle of m 2 41 .What is m 1? How do you know?134Chapter 3hs geo pe 0302.indd 134Parallel and Perpendicular Lines1/19/15 9:23 AM

3.2ExercisesDynamic Solutions available at BigIdeasMath.comVocabulary and Core Concept Check1. WRITING How are the Alternate Interior Angles Theorem (Theorem 3.2) and the Alternate ExteriorAngles Theorem (Theorem 3.3) alike? How are they different?2. WHICH ONE DOESN’T BELONG? Which pair of angle measures does not belong with theother three? Explain.m 1 and m 3m 2 and m 4m 2 and m 3m 1 and m 512435Monitoring Progress and Modeling with Mathematics10.In Exercises 3–6, find m 1 and m 2. Tell whichtheorem you use in each case. (See Example 1.)3.4.117 (8x 6) 150 11225.6.1 2140 122 118 421In Exercises 11 and 12, find m 1, m 2, and m 3.Explain your reasoning.11.1 280 3In Exercises 7–10, find the v

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

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