Chapter 3: Parallel And Perpendicular Lines Guided Notes

Name:Chapter 3:Parallel and Perpendicular LinesGuided NotesGeometryFall Semester

3.1 Identify Pairs of Lines and AnglesTermDefinitionparallel lines(// or )skew linesparallel planesPostulate 13ParallelPostulateIf there is a line and a point not on the line,then there is exactly one line through thepoint parallel to the given line.Postulate 14If there is a line and a point not on the line,Perpendicularthen there is exactly one line through thePostulatepoint perpendicular to the given line.TransversalThe lines the transversal intersects do notneed to be parallel; the transversal can alsobe a ray or line segment.CH.3 Guided Notes, page 2Example

CH.3 Guided Notes, page 3Angles formed by Transversalsexterior anglesinterior anglescorrespondinganglesalternateinterior anglesalternateexterior anglesconsecutiveinterior angles(same-sideinterior angles)Examples:1. Think of each segment in the figure as part of a line. Which line(s) or plane(s)in the figure appear to fit the description?a) Line(s) parallel to AF and containing point E.b) Line(s) skew to AF and containing point E.c) Line(s) perpendicular to AF and containingpoint E.d) Plane(s) parallel to plane FGH and containingpoint E.

CH.3 Guided Notes, page 42. Use the diagram at the right to answer each question.a) Name a pair of parallel lines.b) Name a pair of perpendicular lines.c) Is AB !BC?3. From the diagram, identify all pairs of . . .a) corresponding anglesb) alternate interior anglesc) alternate exterior anglesd) consecutive (same-side) interior angles

3.2 Use Parallel Lines and TransversalsTermCH.3 Guided Notes, page 5DefinitionIf two parallel lines are cut by aPostulate 15Correspondingtransversal, then the pairs of correspondingangles are congruent.Angles PostulateIf two parallel lines are cut by aTheorem 3.1Alternatetransversal, then the pairs of alternateinterior angles are congruent.Interior AnglesTheoremIf two parallel lines are cut by aTheorem 3.2Alternatetransversal, then the pairs of alternateexterior angles are congruent.Exterior AnglesTheoremIf two parallel lines are cut by aTheorem 3.3Consecutivetransversal, then the pairs of same sideinterior angles are supplementary.Interior AnglesTheoremExamples:1. Given the diagram at right, which numbered angles havea measure of 125 ?Example

CH.3 Guided Notes, page 62. Find the value of x.3. A taxiway is being constructed that intersects two parallel runways at anairport. You know that m!2 98 . What is m!1 ? How do you know?

3.3 Prove Lines are ParallelTermDefinitionIf two lines are cut by a transversal so thePostulate 16Correspondingcorresponding angles are congruent, then thelines are parallel.Angles ConverseIf two lines are cut by a transversal so theTheorem 3.4Alternatealternate interior angles are congruent, thenthe lines are parallel.Interior AnglesConverseIf two lines are cut by a transversal so theTheorem 3.5Alternatealternate exterior angles are congruent, thenthe lines are parallel.Exterior AnglesConverseIf two lines are cut by a transversal so theTheorem 3.6Consecutiveconsecutive interior angles aresupplementary, then the lines are parallel.Interior AnglesConverseparagraph proofIf two lines are parallel to the same line,Theorem 3.7TransitiveProperty ofParallel Linesthen they are parallel to each other.CH.3 Guided Notes, page 7Example

CH.3 Guided Notes, page 8Examples:1. Find the value of x that makes m // n .2. How can you tell whether the sides of the flag of Nepal are parallel?3. Write a paragraph proof. In the figure, a // b and !1 is congruent to !3.Prove x // y .Plan:Proof:

CH.3 Guided Notes, page 94. Each utility pole shown is parallel to the pole immediately to its right. Use theTransitive Property of Parallel Lines to explain why the leftmost pole is parallel to therightmost pole.

3.4 Find and Use Slopes of LinesTermDefinitionslopepositive slopenegative slopezero slope(slope of zero)(no slope)A horizontal line.undefined slopeA vertical line.In a coordinate plane, two nonvertical linesPostulate 17Slopes ofare parallel if and only if they have thesame slope.Parallel LinesAny two vertical lines are parallel.In a coordinate plane, two nonvertical linesare perpendicular if and only if the productPostulate 18of their slopes is -1.Slopes ofPerpendicularLinesThe slopes of the two lines that areperpendicular are negative reciprocals ofeach other. Horizontal lines areperpendicular to vertical lines.“if and only if”form(iff)The form used when both a conditional andits converse are true.CH.3 Guided Notes, page 10Example

CH.3 Guided Notes, page 11Examples:1. Find the slope of line a and line c.2. Find the slope of each line. Which lines are parallel?3. Line h passes through points (1,-2) and (5,6). Graph the line perpendicular to h thatpasses through the point (5,2).

CH.3 Guided Notes, page 124.Analyze the graph. A trucker made three deliveries.The graph shows the trucker’s distance to thedestination from the starting time to the arrivaltime for each delivery. Use the slopes of the linesto make a statement about the deliveries.

3.5 Write and Graph Equations of LinesTermDefinitionslope-interceptformstandard formx-intercepty-interceptExamples:1. Write an equation of the line in slope-intercept form.CH.3 Guided Notes, page 13Example

CH.3 Guided Notes, page 142. Write an equation of the line passing through the point (1,-1) that is parallel to the linewith the equation y 2x !1 .3. Write an equation of the line passing through the point (3,2) that is perpendicular tothe line with the equation y !3x 1.4. The graph at right models the total cost of renting an apartment. Write an equationof the line. Explain the meaning of the slope and the y-intercept of the line.

CH.3 Guided Notes, page 155. Graph 2x 3y 6 . The equation is in STANDARD FORM, so use the intercepts.6. Solve a real world problem. You can buy a magazine at a store for 3. You cansubscribe yearly to the magazine for a flat fee of 18. After how many magazines is thesubscription a better buy?

CH.3 Guided Notes, page 163.6 Prove Theorems about Perpendicular LinesTermDefinitionIf two lines intersect to form a linear pairTheorem 3.8of congruent angles, then the lines areperpendicular.If two lines are perpendicular, then theyTheorem 3.9intersect to form four right angles.If two sides of two adjacent acute anglesTheorem 3.10are perpendicular, then the angles arecomplementary.If a transversal is perpendicular to one ofTheorem 3.11two parallel lines, then it is perpendicular toPerpendicularthe other.TransversalTheoremIn a plane, if two lines are perpendicular toTheorem 3.12LinesPerpendicular toa TransversalTheoremdistance from apoint to a linedistancebetween twoparallel linesthe same line, then they are parallel to eachother.Example

CH.3 Guided Notes, page 17Examples:1.2.In the diagram at right, !1 " !2 . What can you conclude about a and b ?In the diagram at right, !1 " !2 . Prove that !3 and !4 are complementary.Given:Prove:StatementsReasons1. !1 " !21.2.2. lin.pr.,! "s # 3. !3 and !4 are complementary.3.

CH.3 Guided Notes, page 183. Determine which lines, if any, are parallel in the diagram. Explain.

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