Chapter 3.1

Parallel Lines and Transversals

Relationships Between Lines and Planes

Parallel Lines – The old definition was “lines that didn’t intersect”

This definition is now not good enough.

l

m

Are lines l and m parallel?

What more do we need in the def?

Parallel Lines

Parallel Lines are coplanar lines that do not intersect.

The symbol of parallel is this….||.

On diagrams we can use something like this to indicate that the lines are parallel.

Parallel Planes

Planes can be parallel.

Planes don’t have to be coplanar to be parallel. As a matter of fact, if planes are coplanar then they are the same plane.

Example of parallel planes.

A

B C

D

E

F G

H

Plane AEHD is parallel to PlaneBFGC.

Any others?

Skew Lines

Skew lines are lines that are not coplanar.

These lines can never be parallel.Because they have to be coplanar to be parallel.

They can never intersect.Because they have to be coplanar to intersect too.

To find Skew lines you need to find noncoplanar lines.

To do this you could eliminate all…Parallel Lines

Coplanar Lines

Skew Lines

A

B C

D

E

F G

H

Find all lines skew to line AE.1) Find all lines parallel to line AE. BF DH CG

������������������������������������������������������������������ �����

2) Eliminate all coplanar lines to line AE.All remaining lines are Skew to line AE.

Another Example

FGEHAD

A

B C

D

E

F G

H

Find all lines skew to line BC.1) Find lines parallel to line BC

2) Find lines coplanar to line BC CDABBFCG

All remaining lines are skew to line BC!

Lines DH, AE, EF and GH are skew to line BC!

Transversals

Transversal – A transversal is line that intersects two or more coplanar lines in different places.

Notice the word parallel is not in the definition.

When a transversal intersects two coplanar lines in two different places it creates EIGHT different angles.

Transversals (Con’t)

Two coplanar lines. l

m

A transversal.

n

Eight angles.

1 2

3 4

675

8

Think of lines l and m as the rails of a railroad track and line n as the railroad tie.

If you do, then you will see that the “choo-choo train will travel right or left on the rails.

Transversals (Con’t)

l

m

1 2

3 4

675

8

n

Thinking of the choo-choo train,which angles are interior of the “rails”?

<3, <4, <5 & <6

Which angles are exterior of the “rails”?

<1, <2, <7 & <8Important angle pairs of angles made by a transversal intersecting a pair of lines.

They are Alternate Interior, Alternate Exterior, ConsecutiveInterior, Consecutive Exterior and Corresponding.

Transversal and Angle Pairs

Alternate Interior Angle Pairs –A pair of interior angles, one per vertex, that are

on alternate sides of the transversal.Alternate Exterior Angle Pairs –

A pair of exterior angles, one per vertex, that are on alternate sides of the transversal.

Consecutive Interior Angle Pairs – A pair of interior angles, one per vertex, that are

on the same side of the transversal.Consecutive Exterior Angle Pairs –

A pair of exterior angles, one per vertex, that are on the same side of the transversal.

Corresponding Angle Pairs – A pair of angles that “sit on top of each other”

when one vertex is put on top of the other.

Keys to this:Do you see the word parallel in any of the

definitions?

The angle pairs are one per vertex – meaning they are not adjacent, nor linear pair.

Transversal and Angle Pairs

Transversals and Angle Pairs

l

m

1 2

3 4

675

8

n

Alternate Interior

<3 & <6<4 & <5

Alternate Exterior

<1 & <8<2 & <7

Consecutive Exterior

<1 & <7<2 & <8

Consecutive Interior

<3 & <5<4 & <6

l

m

1 2

3 4

675

8

n

Transversals and Angle Pairs

Corresponding Angles

<1 & <5

<2 & <6

<3 & <7

<4 & <8

All the angle pairs have been made by the transversal, line n, intersecting two lines, l and m in two different places.

Chapter 3.2

Angles and Parallel Lines

Now, parallel lines!

We made a big deal last section that the angles are created by a transversal cutting two different lines in two different locations.What happens when the transversal cuts

parallel lines?What happens is that the five named

relationships we studied now have specific relationships.

Cutting Parallel Lines

When a transversal cuts parallel lines, the five pairs of angles have specific relationships.Alternate Interior Angles (Alt Int) are

congruent.Alternate Exterior Angles (Alt Ext) are

congruent.Corresponding Angles (Corr) are

congruent.

Cutting Parallel Lines (Con’t)

Consecutive Interior Angles (Con Int) are supplementary.

Consecutive Exterior Angles (Con Ext) are supplementary.

These relations hold true ONLY if the transversal cuts parallel lines.

If the lines are not parallel, then the relationships don’t exist.

Example

< Pairs Trans Lines Name Equation

n

l pm

m||n1 2 3 45 6 7 8

9 10 11 1213 14 15 16

<1 & <9

<6 & <3<16 & <3

<16 & <4

ln

p

p

m, n

m, n

l, p

m, n

Corr m<1 = m<9

Alt Int None

Alt Ext m<16 = m<3

Con Ext m<16+m<4=180

Perpendicular Transversals

Perpendicular Transversal Theorem – In a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other line.

Auxiliary Line

An Auxiliary Line is a line that you can draw any where you like. You can draw them parallel or perpendicular or not.

30°

45°

?45°30° Using the

AAP, you get m<75°

Chapter 3.3

Slopes of Lines

Slope

The slope of a line is the ratio of the vertical rise to it’s horizontal run.Slope = Rise

Run

m = (y2 – y1)/(x2 – x1)Slopes can tell you if two lines are parallel

(||), perpendicular ( | ) or neither.Slopes can not tell if two lines are

congruent or not.

Slopes of Lines

4

2

-2

-5 5

A

C

Create a right triangle

Find vertical change

Find horizontal change

3

7

m= 3/7 7

3

)4(3

)1(2

m

Four Types of Slopes

4

2

-2

-5 5

Positive slope – line going “uphill”

Negative slope – line going “downhill”

Zero slope – there is “no hill”Undefined Slope – there is a “wall”

Parallel vs. Perpendicular

Slopes can tell you if a pair of lines are parallel, perpendicular or neither.Parallel lines are lines that have the same

slope.If the slope of one line is 3/2 and the

slope of the other line is also 3/2 then the two lines are parallel.Slopes of zero are parallel.Undefined slopes are parallel.

Parallel vs. Perpendicular

For two lines to be perpendicular the slopes have to be “opposite signed, reciprocals”.

This is pretty easy, if one line has a slope of m = 1/3 then the slope of the perpendicular line must be m = -3/1.

The only thing that is unusual is that if a line has a slope of zero then the slope of the perpendicular line is undefined.

The reverse is also true.

Chapter 3.4

Equations of Lines

Equations of Lines

There are three forms of a line equation that you should be familiar with.

Slope Intercept Form y = mx + b.

Standard Form Ax + By = C

Point Slope Form y – y1 = m(x – x1)

It doesn’t matter which form you use, you should be able to move from one form to the other simply.

Slope Intercept Form

Probably the most common form of a line equation is the Slope Intercept Form.

The reason why this is the most common form is once you have the line equation in this form you can simply read the slope and the y intercept.

You have y = mx + b, where m = slope of the line and b is the y intercept.

The coordinate of the y intercept is always (0, b)

Standard Form

The standard form of the line equation is Ax + By = C.Both the x and y are on one side of the

equal sign, while the constant is on the other side.A, B and C are integers (no fractions or

decimals) and A must be positive.Here you can’t just read the slope and the

y intercept, you will need to solve for them.

Point Slope Form

The point slope form may have been the first form a linear equation that you learned.

The equation is y – y1 = m(x – x1), where m is the slope and the point that the slope goes through is (x1, y1).

To use this equation you will need to plug in m for the slope and (x1, y1) the point.

Example – Slope Intercept FormGiven a point and slope

Find the equation of a line with a slope of 2/3 through the point (3, -4)Substitute m= 2/3, x = 3 and y = -4 into the

slope intercept form of the line equation.You get -4 = (2/3)3 + bThen solve for b.b = -6Substitute m and b back into the equation.y = (2/3)x - 6

Example – Slope Intercept FormGiven two points

Find the equation of a line through (-3, -1) and (6, 5)

First you need to find the slope.m = (-1 – 5)/(-3 – 6) = -6/-9 = 2/3.

Now pick one point and do what we just did.

-1=(2/3)(-3) + b solving for b

b = 1

So y=(2/3)x + 1

Another Example

Find the equation of the perpendicular line through the line connecting (3, 5) and ( 6, -2) through (-1, 4)

First, find the slope between the two points m=(-2 – 5)/(6 – 3) = -7/3

Second, find the slope of the line perpendicular to the line with a slope of -7/3, so m = 3/7

Next, solve for b: 4 = (3/7)(-1) + b, so b = 31/7

So, y = (3/7)x + 31/7 is the equation of the perpendicular line.

Chapter 3.5

Proving Lines Parallel

Review from 3.2

If a transversal cuts parallel lines, then…alternate interior angle pairs are congruent.

alternate exterior angle pairs are congruent.

consecutive interior angle pairs are supplementary.

consecutive exterior angle pairs are supplementary.

corresponding angle pairs are congruent.

Now let us work on the converses of these statements.

Converses

If Alternate Interior Angle pairs are congruent…

If Alternate Exterior Angle pairs are congruent…

If Consecutive Interior Angle pairs are supplementary…

If Consecutive Exterior Angle pairs are supplementary…

If Corresponding Angle pairs are congruent..

…then the transversal intersects parallel lines.

Examples

n

l

m

p30°

30°

n & p are || - Alt Int < Thrm

30°l & m are || - Corr < Thrm

30°

n & p are || - Alt Ext < Thrm30°

30°150°

l & m are || - Con Int < Thrm

Another Example

3x + 12°

x + 5°

Find the value of x to make these two lines parallel.

What kind of angles are these? Con Ext

What do Con Ext anglesneed to be to make linesparallel? Supplementary

(3x + 12) + (x + 5) = 1804x + 17 = 180

4x = 163 x = 163/4

Chapter 3.6

Perpendiculars and Distance

Distance Between a Point and a Line

The distance between a point and a line you will need to find the length of the perpendicular segment from the point to that line.

Distance from a Point to a Line

A

B

C

DFind the distance between:

B and line AD….

D and line AB

D and line CB

A and line CB

B and line CD

Distance Between Parallel Lines

To find the distance between two parallel lines, just pick a point on one of the lines and find the distance between the point and a line.Alternate definition of Parallel Lines – Coplanar Lines that are Equidistant.

Parallel Line Theorem

In a plane, if two lines are equidistant from a third line, then the two lines are parallel to each other.

This is not an example of the transitive postulate.

Example – Coordinate Geo

4

2

-2

-4

-5 5

C

B A

Find the distance fromC to line AB.

Construct a segment perpendicular from C to line AB.

Hint: If the line is horizontal or vertical, then just count the boxes.

So, the distance between C and line AB is 6.

Example – Coordinate Geo (H)

4

2

-2

-4

-5 5

C

B A

Find the distance from A to BC.Since BC is not Horor Vert, we need to do a lot of work.First, find the slope of line BC. m = 3/1

Third, find the perpendicular slope. m = -1/3Fourth, find the equation of the perpendicular line fromA to BC. y = (-1/3)x - 1

D

Second, find equation of BC y = 3x + 13

Example - Continued

So, now we have the equation of BC y = 3x + 13 & the equation of AD y = (-1/3)x – 1.

Since both equations are written in slope intercept form you can set them equal to each other.

3x + 13 = (-1/3)x – 1….. mult. by 3

9x + 39 = -x – 3

10x = -42 x = - 4.2 and y = .4

Now, find the distance between A ( 3, -2) and D ( - 4.2, 0.4)

Non-Euclidean Geometry (H)

Euclidean Geometry is what we are doing up to this date. It is the geometry of planes.

Non-Euclidean Geometry is the geometry of other types of systems such as spheres or cones…..

In Spherical Geometry, we are dealing with great circles, circles etc.

See page 165…