chapter 3. 3.1 lines and angles first thing we’re going to do is travel to another dimension

Parallel and Perpendicular Lines

Parallel and Perpendicular LinesChapter 33.1

Lines and AnglesFirst thing were going to do is travel to another dimension

THE THIRD DIMENSIONFirst thing were going to do is travel to another dimension

Once we get there well discuss Parallel, Perpendicular, and Skew lines

First thing were going to do is travel to another dimensionDiagramed

Parallel linesDiagramed

Parallel linesCoplanar lines that dont intersect.

Parallel linesAB, CF, EG, DJDiagramed

Parallel linesAB, CF, EG, DJAD, BJ, FG, CEDiagramed

Perpendicular linesDiagramed

Perpendicular linesIntersect to make a right angle

Perpendicular linesAB and BJIntersect to make a right angle

Perpendicular linesAB and BJAB and BCBJ and BCIntersect to make a right angle

Perpendicular linesAB and BJAB and BCBJ and BCIf 2 lines are perpendicular to the same line, are they perpendicular to each other?

Skew linesSomething perhaps thats gnu

Skew linesSomething perhaps thats gnu

Skew linesDefines as lines in different planes that are not parallel.

Skew linesThe only reason they dont intersect is because they are not coplanar.

Skew linesExamples:

Skew linesAB and EJExamples:

Skew linesAB and EJJD and FGExamples:

Skew linesAB and EJJD and FGDG and CEExamples:

Make sure that

Given a diagram:Make sure thatGiven a diagram:-Identify the relationship between a pair of linesMake sure thatGiven a diagram:-Identify the relationship between a pair of lines.-Label lines so that the desired relationship is shownMake sure thatGiven a diagram:-Identify the relationship between a pair of lines.-Label lines so that the desired relationship is shownComplete the Got It? on page 141Use above it as a guide if you desire.Complete the Got It? on page 141Returning to the flat world

On a plane, when lines intersect two or more lines at distinct points, the angles formed at these points create special angle pairs.Returning to the flat world

Their description and location is based upon a transversal.Returning to the flat world

Their description and location is based upon a transversal.

A line that intersects two or more lines at distinct points.Returning to the flat worldThese will break down into interior and exterior locations.Page 141 in your book.These will break down into interior and exterior locations.Page 141 in your book.-Interior angles are found between the 2 lines that are intersectedThese will break down into interior and exterior locations.Page 141 in your book.-Interior angles are found between the 2 lines that are intersected-As you can guess, exterior angles are then found outside these same lines.These will break down into interior and exterior locations.Now we throw in alternate which involves opposite sides of the transversal.Page 142Now we throw in alternate which involves opposite sides of the transversal.Names and DescriptionsThe glowing line is the transversal.

Names and DescriptionsAlternate interior angles are nonadjacent interior angles found on opposite sides of the transversal.

3 and 6Alternate interior angles are nonadjacent interior angles found on opposite sides of the transversal.

4 and 5Alternate interior angles are nonadjacent interior angles found on opposite sides of the transversal.

Names and DescriptionsAlternate exterior angles are nonadjacent exterior angles found on opposite sides of the transversal.

1 and 8Alternate exterior angles are nonadjacent exterior angles found on opposite sides of the transversal.

2 and 7Alternate exterior angles are nonadjacent exterior angles found on opposite sides of the transversal.

Names and DescriptionsSame-side interior angles are nonadjacent angles that line on the same side of the transversal.

3 and 5Same-side interior angles are nonadjacent angles that line on the same side of the transversal.

4 and 6Same-side interior angles are nonadjacent angles that line on the same side of the transversal.

Names and DescriptionsCorresponding angles are angles found on the same side of the transversal in the same corresponding or relative position.

1 and 5Corresponding angles are angles found on the same side of the transversal in the same corresponding or relative position.




HomeworkPage 144 145

11 24, 30 35, 37 42 Answer the questions, identify the desired relationships.Properties of and Proving Parallel Lines3.2 3.3Heres what youre going to do

1)On a sheet of notebook, darken in 2 horizontal lines a few inches apart.Heres what youre going to do

1)On a sheet of notebook, darken in 2 horizontal lines a few inches apart.2)Create a transversal thatis not perpendicular to your 2 lines.Heres what youre going to do

2)Create a transversal thatis not perpendicular to your 2 lines.3)Measure all 8 angles thatare formed by the trans-versal and the lines youdarkened.Heres what youre going to do

Now for the thought process:What is special about the lines you darkened?Now for the thought process:What is special about the lines you darkened?They are parallelNow for the thought process:What is special about the lines you darkened?They are parallelWhat is special about pairs of angles you measured?Now for the thought process:What is special about the lines you darkened?They are parallelWhat is special about pairs of angles you measured?They are congruentNow for the thought process:This is not a coincidence

If a transversal intersects 2 parallel lines:This is not a coincidenceIf a transversal intersects 2 parallel lines:(1)Alternate interior angles are congruent.This is not a coincidenceIf a transversal intersects 2 parallel lines:(1)Alternate interior angles are congruent.(2)Alternate exterior angles are congruent.This is not a coincidenceIf a transversal intersects 2 parallel lines:(2)Alternate exterior angles are congruent.(3)Corresponding angles are congruent.This is not a coincidenceIf a transversal intersects 2 parallel lines:(3)Corresponding angles are congruent.(4)Same side interior angles are supplementary.This is not a coincidenceA postulate

3.1If a transversal intersects two parallel lines, then same side interior angles are supplementaryA postulate

3.1If a transversal intersects two parallel lines, then same side interior angles are supplementaryA list of theorems

3.1If a transversal intersects two parallel lines, then alternate interior angles are congruentA list of theorems

3.2If a transversal intersects two parallel lines, then corresponding angles are congruentA list of theorems

3.3If a transversal intersects two parallel lines, then alternate exterior angles are congruent.A list of theorems

The long way to find angle measures. 12 3 4

56 7 8Let m3 = 82The long way to find angle measures. 12 3 4

56 7 8Let m3 = 82m2 = ____m1 = ____m4 = ____Vertical angle conjecture 12 3 4

56 7 8Let m3 = 82m2 = 82m1 = ____m4 = ____Linear Pair Angle Conjecutre 12 3 4

56 7 8Let m3 = 82m2 = 82m1 = 98m4 = ____Linear Pair or Vertical Angle Conjecture 12 3 4

56 7 8Let m3 = 82m2 = 82m1 = 98m4 = 82Now we march on to the other point of intersection 12 3 4

56 7 8Let m3 = 82m2 = 82m1 = 98m4 = 82Now we march on to the other point of intersection 12 3 4

56 7 8Let m3 = 82m5 = ____m6 = ____m7 =____m8 =____By the Same-Side Conjecture 12 3 4

56 7 8Let m3 = 82m5 = ____m6 = ____m7 =____m8 =____By the Same-Side Conjecture 12 3 4

56 7 8Let m3 = 82m5 = 98m6 = ____m7 =____m8 =____By the Linear Pair Conjecture 12 3 4

56 7 8Let m3 = 82m5 = 98m6 = 82m7 =____m8 =____By the Linear Pair Conjecture 12 3 4

56 7 8Let m3 = 82m5 = 98m6 = 82m7 =____m8 =____What is the defined relationship between 3 and 6?Alternate Interior Angles!!! 12 3 4

56 7 8Let m3 = 82m5 = 98m6 = 82m7 =____m8 =____What is the defined relationship between 3 and 6?Alternate Interior Angles!!! 12 3 4

56 7 8Let m3 = 82m5 = 98m6 = 82m7 =82m8 =____Which has a corresponding angle relationship with 3. 12 3 4

56 7 8Let m3 = 82m5 = 98m6 = 82m7 = 82m8 =____Which has a corresponding angle relationship with 3. 12 3 4

56 7 8Let m3 = 82m5 = 98m6 = 82m7 = 82m8 = 98Which makes alternate exterior angle magic with 1. 12 3 4

56 7 8Let m3 = 82m5 = 98m6 = 82m7 = 82m8 = 98Now the short methodNow the short methodIf youre asked to find, not justify or prove that angles are congruent or have the same angle measure:Now the short methodIf youre asked to find, not justify or prove that angles are congruent or have the same angle measure:-All acute anglesare .Now the short methodIf youre asked to find, not justify or prove that angles are congruent or have the same angle measure:All acute angles are .All obtuse angles are

Now the short methodIf youre asked to find, not justify or prove that angles are congruent or have the same angle measure:All acute angles are .All obtuse angles are The sum of an acute and an obtuse angle = 180

Provided the lines are parallel.If youre asked to find, not justify or prove that angles are congruent or have the same angle measure:All acute angles are .All obtuse angles are The sum of an acute and an obtuse angle = 180

Formal proofGiven: j || kProve: 4 6

Formal proof

StatementReasonFormal proof

StatementReasonm3 + m4 = 180Formal proof

StatementReasonm3 + m4 = 180Linear Pair ConjectureFormal proof

StatementReasonm3 + m4 = 180Linear Pair Conjecturem3 + m6 = 180Formal proof

StatementReasonm3 + m4 = 180Linear Pair Conjecturem3 + m6 = 180Same Side Interior Angle ConjectureFormal proof

StatementReasonm3 + m4 = 180Linear Pair Conjecturem3 + m6 = 180Same Side Interior Angle Conjecturem3 + m4 = m3 + m6 Transitive PropertyFormal proof

StatementReasonm3 + m6 = 180Same Side Interior Angle Conjecturem3 + m4 = m3 + m6 Transitive Propertym4 = m6Subtraction Property of EqualityFormal proof

StatementReasonm3 + m4 = m3 + m6 Transitive Propertym4 = m6Subtraction Property of Equality4 6

Formal proof


Definition of CongruenceYou will most likely have to do one of these on your next quiz.


Definition of CongruenceIf it does ask you to justify

If it does ask you to justifyInclude a definition or theorem that allows you to state your angle relationship.If it does ask you to justifyInclude a definition or theorem that allows you to state your angle relationship.3.2 Practice # 1Why is 3 also 132?Include a definition or theorem that allows you to state your angle relationship.3.2 Practice # 1Why is 3 also 132?Include a definition or theorem that allows you to state your angle relationship.3.2 Practice # 1

3:Vertical AnglesWhy is 3 also 132?Include a definition or theorem that allows you to state your angle relationship.3.2 Practice # 1

3:Vertical Angles5:Why is 3 also 132?Include a definition or theorem that allows you to state your angle relationship.3.2 Practice # 1

3:Vertical Angles5: CorrespondingWhy is 3 also 132?Include a definition or theorem that allows you to state your angle relationship.3.2 Practice # 1

3:Vertical Angles5: Corresponding7:Why is 3 also 132?Include a definition or theorem that allows you to state your angle relationship.3.2 Practice # 1

3:Vertical Angles5: Corresponding7:Alternate ExteriorYou try #2Include a definition or theorem that allows you to state your angle relationship.3.2 Practice # 1

3:Vertical Angles5: Corresponding7:Alternate ExteriorSolution5 is 78 because of alternate interior angles.Solution5 is 78 because of alternate interior angles.

1 is 78 because of vertical angles.Be specific!!!1 is 78 because of vertical angles.

7 is 78 because of corresponding anglesBe specific!!!1 is 78 because of vertical angles.

7 is 78 because of corresponding anglesAlternative:

7 makes a vertical angle pair with #5If you dont write anything, we assume you are talking about the angle measure given to you.1 is 78 because of vertical angles.

7 is 78 because of corresponding anglesAlternative:

7 makes a vertical angle pair with #5Similar idea, moving to #5130 is the reference angle.Similar idea, moving to #5130 is the reference angle.Angle 1 is _____ because it makes a __________ ________ with the 130 angle.Similar idea, moving to #5130 is the reference angle.Angle 1 is 50 because it makes a linear pair with the 130 angle.Similar idea, moving to #5130 is the reference angle.Angle 1 is 50 because it makes a linear pair with the 130 angle.Angle 2 is Similar idea, moving to #5130 is the reference angle.Angle 1 is 50 because it makes a linear pair with the 130 angle.Angle 2 is 130 because it is a corresponding angle to the 130.Similar idea, moving to #5130 is the reference angle.Angle 1 is 50 because it makes a linear pair with the 130 angle.Angle 2 is 130 because it is a corresponding angle to the 130.You do #6Things to remember in sketches:

Make sure their exists a relationship between the angles.Things to remember in sketches:

Make sure their exists a relationship between the angles.Touch the same transversal, and that the lines are parallel.Things to remember in sketches:

Make sure their exists a relationship between the angles.Touch the same transversal, and that the lines are parallel.Keep this in mind as we tackle the remaining problems.Things to remember in sketches:

3.3Proving Lines ParallelReversing the process

It seems weve gone down this road before

Theorem 3-4

Theorem 3-4If 2 lines and a transversal form corresponding angles that are congruent, then the lines are parallel

Theorem 3-5If 2 lines and a transversal form alternate interior angles that are congruent, then the lines are parallel

Theorem 3-6If 2 lines and a transversal form same side interior angles that are supplementary, then the lines are parallel

Theorem 3-7If 2 lines and a transversal form alternate exterior angles that are congruent, then the lines are parallel

1 6

Lets use the 3.3 Practice to see how to problem solve1 6 (A)Find the congruent anglesLets use the 3.3 Practice to see how to problem solve

1 6 (A)Find the congruent angles(B)Determine the lines they areon.Lets use the 3.3 Practice to see how to problem solve

1 6 (A)Find the congruent angles(B)Determine the lines they areon.NOT THE TRANSVERSAL!!!

1 6 (A)Find the congruent angles(B)Determine the lines they areon.(C)Identify the relationship between them to justify.Lets use the 3.3 Practice to see how to problem solve7:Poof A proofLets use the 3.3 Practice to see how to problem solve8: A walk through

Lets use the 3.3 Practice to see how to problem solve9 14

Lets use the 3.3 Practice to see how to problem solve9 14(A)Work under the belief thatthe lines are parallel.

Lets use the 3.3 Practice to see how to problem solve9 14(A)Work under the belief thatthe lines are parallel. (B)Identify the relationship and set up an equation.

Lets use the 3.3 Practice to see how to problem solve9 14(A)Work under the belief thatthe lines are parallel. (B)Identify the relationship and set up an equation.

This will be either congruent or supplementary, if possible.15 20: Some of my faves(1)Look at the angle pair they provideyou.15 20: Some of my faves(1)Look at the angle pair they provideyou.(2)Identify the relationship, if any, from the diagram.15 20: Some of my faves(1)Look at the angle pair they provideyou.(2)Identify the relationship, if any, from the diagram.(3)Find the desired value.15 20: Some of my faves#RelationshipJustification15 20: Some of my faves#RelationshipJustification1511 & 10 are supplementary15 20: Some of my faves#RelationshipJustification1511 & 10 are supplementaryLines u and t are parallel because same side interior angles are supplementary.15 20: Some of my faves#RelationshipJustification1511 & 10 are supplementaryLines u and t are parallel because same side interior angles are supplementary.166 915 20: Some of my faves#RelationshipJustification166 9Lines a and b are parallel because alternate interior angles are congruent.15 20: Some of my faves#RelationshipJustification166 9Lines a and b are parallel because alternate interior angles are congruent.You fill in the rest#RelationshipJustification1713 and 14 supplementaryNothing: this is always true no matter what lines are parallel.1813 and 15 are congruentLines t and u are parallel because corresponding angles are congruent1912 is supplementary to 33 is also supplementary to 4 because of linear pairs. By the congruent supplements theorem, 4 and 12 are congruent, which are corresponding angles, making a and b parallelYou fill in the rest#RelationshipJustification1912 is supplementary to 33 is also supplementary to 4 because of linear pairs. By the congruent supplements theorem, 4 and 12 are congruent, which are corresponding angles, making a and b parallel202 and 13 are congruenta and b are parallel since alternate exterior angles are congruent.You fill in the rest

chapter 3. 3.1 lines and angles first thing we’re going to do is travel to another dimension

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