Geometry Chapter 3 IPad Student - Mrs. Sowatsky's Math

Chapter 3Parallel and PerpendicularLines

3.1 Identify Pairs of Lines and AnglesObjective:Identify angle pairs formed by threeintersecting lines.Essential Question:What angle pairs are formed bytransversals?

New in this Section: So far we have looked at angle pairsformed by 2 linesVertical angles Supplementary angles Complementary angles Linear pairs Now we will look at angle pairs formedby 3 linesInterior angles Same side (consecutive) Alternate interior Exterior angles Corresponding angles

Euclidean Geometry High School Geometry, invented by a Greekmathematician Euclid is based on 5principals known as postulates Weknow them, since they are easyA line can be drawn between 2 points Any line segment can be a line Circles exist with a given radius All right angles are congruent Parallel lines exist In this section we will study the 5thpostulate, presented in our book aspostulate 13

Every line that does not intersect iseither Parallel or Skew! Skew lines do not intersect but do notexist in the same plane Thinkof a cube, the top of the front face and thebottom of therear face Parallel lines are lines that do notintersect and exist in the same plane

Which sides are parallel?Which sides are perpendicular?

Postulate 13(Euclid’s 5th) If there is a line and a point not on the line,then there is exactly one line through thepoint parallel to the given line Ex.L.There is exactly one line through P parallel to linePLineparallel Toline LAny other line isnot parallelL

Perpendicular Postulate Same idea as parallel postulate If there is a line and a point not on the line,then there is exactly one line through the pointperpendicular to the given lineEx. There is exactly one line through P perpendicular to lineL.PLine PerpendicularTo line LAny other line isnot perpendicularL

Symbolic Representations AB CD Translation: line AB is parallel to line CDnmTwo arrow headsare the symbol hereL Translation line n is parallel to line m

Symbolic Representations AB CD Translation: line AB is perpendicular to line CDmThe square is thesymbol heren Translation: line n is perpendicular to line m

In geometry, a line, line segment, or ray thatintersects two or more lines at different points iscalled aA21456l3m8 7BAB is an example of a transversal. It intersectslines l and m.Note all of the different angles formed at the points of intersection.10/6/2015LSowatsky11

Definition of In a plane, a line is a transversal iff itTransversal intersects two or more lines, each at adifferent point.The lines cut by a transversal may or may not be parallel.Parallel Linesbl1 24 31 24 3m5 68 7lmNonparallel Linestb cc5 68 7rt isa transversal for l and m.LSowatskyr is a transversal for b and12c.10/6/2015

Two lines divide the plane into three regions.The region between the lines is referred to as the.The two regions not between the lines is referred toas the .ExteriorInteriorExterior10/6/2015LSowatsky13

When a transversal intersects two lines,angles are formed.These angles are given special names.1 24 3lInterior angles lie between them5 6two lines.8 7tInterior angles areon the opposite sides of thelietransversal.outside the two lines.Interior angles areAlternate Exterioron the same side of theangles are on thetransversal.opposite sides of the14LSowatsky10/6/2015transversal.

When a transversal crosses two lines, the intersectioncreates a number of angles that are related to eachother.Note 1 and 5 below. Although one is an exteriorangle and the other is an interior angle, both lie onthe same side of the transversal.Angle 1 and 5are called.l1 24 3m5 68 7tGive three other pairs of corresponding angles thatare formed:10/6/2015LSowatsky15

Homework:Exercises 3.1Concepts: #1 – 37Regular: # 1 - 42Honors: # 1 – 44

3.2 Use Parallel Lines and TransversalsObjective: Use angles formed by parallellines and transversals.Essential Question: How arecorresponding angles and alternateinterior angles related for two parallellines and a transversal?

If two parallel lines are cut by atransversal, then each pair ofcorresponding angles is .PostulateCorrespondingAngles10/6/2015LSowatsky18

If two parallel lines are cut by atransversal, then each pair ofalternate interior angles is .TheoremAlternateInteriorAngles124 35 68 7 4 610/6/2015LSowatsky 3 519

If two parallel lines are cut by atransversal, then each pair ofconsecutive interior angles is.1 24 3Theorem:ConsecutiveInteriorAngles5 68 7m 4 m 5 180 m 3 m 6 18010/6/2015LSowatsky20

If two parallel lines are cut by atransversal, then each pair ofalternate exterior angles is .AlternateExteriorAnglesTheorem10/6/20151 24 35 68 7 1 7LSowatsky 2 821

s t and c d.sName all the angles that arecongruent to 1.Give a reason for each answer.1591013 14t32671115c4812d16 3 1 6 1 8 1 9 1 14 1 11 9 1 16 14 110/6/2015LSowatsky22

Example 1: Identify congruent anglesThe measure of three of the numberedangles is 120 . Identify the angles. Explainyour reasoning.SOLUTION

Example 2:Find the value of x.SOLUTION

Example 3:Use the diagram.1. If m 1 105 , find m 4, m 5, andm 8. Tell which postulate or theorem youuse in each case.

Example 4:Use the diagram.If3 68 and m 8 (2x 4) ,what is the value of x? Show yoursteps.

Solve a real-world problemWhen sunlight enters a drop of rain, different colors oflight leave the drop at different angles. This process iswhat makes a rainbow. For violet light, m2 40 .What is m1? How do you know?

Homework:Exercises 3.2Concepts: #1 – 38Regular: # 1 – 34, 37, 38Honors: #1 - 41

3.3 Prove Lines are ParallelObjective:Use angle relationships to prove thatlines are parallel.Essential Question:How do you prove lines parallel?

In a plane, if two lines are cut by atransversal so that a pair ofangles is congruent,then the lines are .1Postulate 16aCorrespondingAngles Converse2bIf 1 2,then

Example 1Is there enoughinformation in thediagram to concludethat m n? Explain.ANSWER

In a plane, if two lines are cut by atransversal so that a pairof angles is congruent,Theorem:then the two lines are .AlternateInterioraIf 1 2,Angles1Converse2bthen

In a plane, if two lines are cut by atransversal so that a pair of alternateexterior angles is congruent, then theTheorem: two lines are .Alternate1If 1 2,ExterioraAnglesthenConverseb2

In a plane, if two lines are cut by atransversal so that a pair of consecutiveinterior angles is supplementary, thenTheorem: the two lines are .ConsecutiveInteriorAnglesConverse12aIfm 1 m 2bthen 180,

Example 2:How can you tell whether the sides of thepattern are parallel in the photo of adiamond-back snake?SOLUTION

Example 3Can you prove that lines a and b areparallel? Explain why or why not.

Prove the Alternate Interior Angles ConverseProve that if two lines are cut by a transversal so thealternate interior angles are congruent, then the lines areparallel.GIVEN : 4PROVE :g 5hSTATEMENTS1.4 54.ghREASONS

If two lines are parallel to the same line,then they are parallel to each other.Theorem:TransitivePropertyof ParallelLinesabcIf a b and b c,then

Example: Use the Transitive Property ofParallel LinesThe flag of the United States has 13alternating red and white stripes. Each stripe isparallel to the stripe immediately below it.Explain why the top stripe is parallel to thebottom stripe.

Homework:Exercises 3.3Concepts: #1 – 33Regular: # 1 – 33Honors: # 1 – 33 and choose 2 from 34 37

3.4 Find and Use Slopes of LinesObjective:Find and compare slopes of linesEssential Question:How do you find the slope of a linesgiven the coordinates of two points onthe line?

SlopeThe slope of the nonvertical line passingthrough the points (x1, y1) and (x2, y2) is:y 2 y1 risem x2 x1 run

Example: Find the slope of the line

Example: Find the slope of the line that goesthrough the points (-6, 8) and (3, 4)

Slopes of Parallel and Perpendicular Lines Parallel lines: lines are parallel iff theyhave the same slope.Perpendicular lines: lines areperpendicular iff their slopes arenegative reciprocals of each other.

Example: Tell whether the lines areparallel, perpendicular, or neither.a) Line 1: through (1, -2) and (3, -2)Line 2: through (-5, 4) and (0, 4)b) Line 1: through (-2, -2) and (4, 1)Line 2: through (-3, -3) and (1, 5)

Example: Draw a perpendicular lineLine h passes through (3, 0) and (7, 6).Graph the line perpendicular to h thatpasses through the point (2, 5).

Homework:Exercises 3.4Concepts: Practice 3.4Regular: # 1 – 29, 33 – 35, 39, 40Honors: # 1 – 35, 38, 39, 40

3.5 Write and Graph Equations of LinesObjective:Find equations of linesEssential Question:How do you write an equation of a line?

Slope-Intercept Formy mx b

Example: Write an equation of a line from a graphWrite an equation of the line in slopeintercept form.

Checkpoint: Write an equation of the line inslope-intercept form that passes through (2, -3)and is a) perpendicular to and b) parallel to theline y 2x - 3

Standard FormAx By C

Intercepts A quick way to graph an equation instandard form is to plot its intercepts.

Example: Graph a line with equation instandard formGraph 3x 4y 12.The equation is in standard form, so you canuse the intercepts.

Homework:Exercises 3.5Concepts: Practice 3.5Regular: # 1, 2, 4 – 56 evenHonors: # 1, 2, 4 – 58 even, 61

3.6 Prove Theorems About Perpendicular LinesObjective:Find the distance between a point and alineEssential Question:How do you find the distance between apoint and a line?

Example 1: Draw ConclusionsIn the diagram, ABBC. Whatcan you conclude about1 and2?

Example:Given thatABC ABD,what can you conclude about3 and4? Explain how youknow.

Example:Use the diagram at the right.a) Is b a? Explain yourreasoning.b) Is breasoning.c? Explain your

Distance From a Line The distance from a point to a line is thelength of the perpendicular segment fromthe point to the line. This perpendicularsegment is the shortest distance betweenthe point and the line. For example, thedistance between point A and line k is AB.

Distance From a Line The distance between two parallel lines isthe length of any perpendicular segmentjoining the two lines. For example, thedistance between line p and line m is CD orEF.

Example:a) What is the distance frompoint A to line c?b) What is the distance fromline c to line d?

Homework:Exercises 3.6Concepts: #1 – 22, 26, 29Regular: #1 – 22, 26, 29, 31Honors: # 1 – 22, 26, 29, 31, choose 2from 35 – 38

Chapter 3 Test

Chapter 3 Parallel and Perpendicular Lines . 3.1 Identify Pairs of Lines and Angles Objective: Identify angle pairs formed by three intersecting lines. Essential Question: . Slopes of Parallel and Perpendicular Lines Parallel lines: lines ar