Warm-Up Parallel And Perpendicular Lines

Warm-Up?Parallel and Perpendicular LinesLessonQuestion Lesson GoalsSolve problems involving thebetween lines.Applytransformations toSolve problemsinvolving distanceconstructlines.from a point on aparallel, perpendicular,and skewin three-dimensionalfigures.bisector.W2KWords to KnowFill in this table as you work through the lesson. You may also use the glossaryto help you.lines that lie in the same plane and do not intersectlines that are noncoplanar and do not intersectlines that intersect to form right, or 90-degree, anglesa line, ray, or line segment that intersects a segmentand is perpendicular to the segment at its midpoint Edgenuity, Inc.1

Warm-UpParallel and Perpendicular LinesParallel and Perpendicular LinesParallel lines are coplanar lines thatPerpendicular lines are coplanardo notlines that intersect to form.,or 90 , angles. The word coplanar means that the lines are in the same. A plane in Euclidean geometry is a two-dimensional object that extendsin both of its dimensions and has no height. Edgenuity, Inc.2

InstructionParallel and Perpendicular LinesSlide2The Shortest Distance TheoremShortest distance theorem: The shortestfrom a given lineto a point not on the line is the length of the segmentto the line through the point.Draw segment CD that represents the shortest distance from point C to line AB.CA Edgenuity, Inc.B3

InstructionParallel and Perpendicular LinesSlide3Given: AB and point C not on ABProve: The shortest distance from C to AB is the length of the segment to ABthrough C.CABDStatements1. Draw a segment from C thatintersects AB atReasons1. unique line postulate.2. Draw a different segment, CE, to2. unique line postulate.AB.3. ECD is a right triangle.3. def. of4. CE is the hypotenuse of ECD.4. def. of5. CE is the longest side of ECD,5. def. of hypotenuseso CD .6. The shortest distance from C toAB is Edgenuity, Inc.triangle6. conclusion from step 5.4

InstructionParallel and Perpendicular LinesSlide5The Parallel PostulateParallel postulate: Given a line and a point not on that line, there existsline in the same plane that passes through the givenpoint and isto the given line.Draw the line that passes through the point and is parallel to the line shown.Skew LinesSkew lines are lines that areand do not.In this diagram, planes R and W areRparallel.ED DE and FG arelines. Perpendicular lines are not skewWFGlines, because they're in the same. Parallel lines areskew lines,because they're in the same plane. Edgenuity, Inc.5

InstructionParallel and Perpendicular LinesSlide5Lines in Three DimensionsIdentify any parallel, perpendicular, or skew lines from the diagram.Parallel:B p andlPerpendicular:n and n and lpAmSkew: l and8The Perpendicular Bisector TheoremPerpendicular bisector theorem: The points on the perpendicular bisector,which is a line, ray, or line segment that intersects a segment and isperpendicular to the segment at its midpoint, arefrom the.S QS RE RQE Edgenuity, Inc.6

InstructionParallel and Perpendicular LinesSlide8Proving the Perpendicular Bisector TheoremGiven: P is a point on the perpendicularbisector, l, of MN.lProve: PM PNPBy the unique line postulate, you can drawonly one segment,. Using theMdefinition of reflection, reflect PM over l. Bythe definition ofimage of itself andN, P is theis the image of M.Since reflections preserve length, PM Edgenuity, Inc.Q.7

InstructionParallel and Perpendicular LinesSlide10Applying the Perpendicular Bisector TheoremWhat is the length of segment AC?A Line l is a4𝑥 5bisector of line segment AB. Point C isCDl3𝑥 2from the endpoints of line segment AB. AC B4𝑥 5 3𝑥 2 2𝑥 AC 4(7) 5 BC 2312Converse of the Perpendicular Bisector TheoremConverse of the perpendicular bisector theorem: If a point isfrom the endpoints of a segment, then it is on thebisector of the segment.If this point is equidistantlfrom M and N, then it liesPon the perpendicularbisector of MN.M Edgenuity, Inc.QN8

Summary?Parallel and Perpendicular LinesLessonQuestionWhat special relationships exist between two lines or a line segmentand a line?AnswerSlideReview: Key Concepts2PerpendicularLinesSame planeParallel LinesSame planeSkew Lines Intersect atanglesintersectplaneNever intersectdistance theorem If you have a line and a point that's not on the line, then the shortestdistance between the point and the line is the length of the linesegment which connects the two and that isto the given line. Edgenuity, Inc.9

SummaryParallel and Perpendicular LinesSlide2 Parallel postulate If you have a line and a point that's not on the line, then there existsexactly one line that goes through that point and istothe given line. Perpendicular bisector theorem Every point along the perpendicularof a line segmentis equidistant from the line segments endpoints. Converse of the perpendicular bisector theorem If a point isfrom the endpoints of a line segment,then the point must be on the line segment's perpendicular bisector. Edgenuity, Inc.10

SummaryParallel and Perpendicular LinesUse this space to write any questions or thoughts about this lesson. Edgenuity, Inc.11

Skew Lines Skew lines are lines that are and do not . In this diagram, planes R and W are parallel. DEand FGare lines. Perpendicular lines are not skew lines, because they're in the same . Parallel lines are skew lines,