chapter 3 notes. 3.1 lines and angles two lines are parallel if they are coplanar and do not...

Chapter 3 Notes

3.1 Lines and Angles

Two lines are PARALLEL if they are COPLANAR and do not INTERSECT

Two lines are SKEW if they are NOT COPLANAR and do not INTERSECT

D

A

C

B

G

E

H

FArrows on line mean they are parallel

Two planes that do not Intersect are called PARALLEL planes.

D

A

C

B

G

E

H

F

A line and a plane are parallel if they do not intersect.

Line segments and rays can be parallel too!

TY

RO

TY

R O

As long as the lines going through them are also parallel.

TY

RO

TY

RO

D

A

C

B

G

E

H

F

Let’s name some parallel planes, lines, and some skew lines.

Parallel line postulate If there is a line and a point not on the line, there is EXACTLY one parallel line through the

given point.

Perpendicular line postulate If there is a line and a point not on the line, there is EXACTLY one perpendicular line

through the given point.

Not Parallel

1 2

5 6

A ______________ is a line that INTERSECTS two or more COPLANAR lines at different points.

Two angles are ________________ ANGLES if they occupy ________________ positions.Two angles are ________________ __________________ if they LIE _________ the two lines on __________ sides of the ________________.Two angles are _____________________________ if they LIE __________ the two lines on ___________ sides of the TRANSVERSAL.

Two angles are ______________________________ (also called same side interior angles) if they LIE _________ the two lines on the _______ sides of the TRANSVERSAL.

3 4

7 8

Given a point off a line, draw a line perpendicular to line from given point.1) From the given point, pick any arc and mark the circle left and right.

2) Those two marks are your endpoints, and construct a perpendicular bisector just like the previous slide.

3.3 – Parallel Lines and Transversals

Corresponding Angles Postulate (CAP)

If two lines cut by transversal are ||, then the corresponding angles are congruent

m

n

1

2

m tochange to

s' def use toneed Note,

21

,||

then

nmIf

m

n

1

4

32

5

Alternate Interior Angles Theorem (AIA Thrm)

If two lines cut by transversal are ||, then the alternate interior angles are congruent

Consecutive Interior Angles Theorem (CIA Thrm)

If two lines cut by transversal are ||, then the consecutive interior angles are supplementary

Alternate Exterior Angles Theorem (AEA Thrm)

If two lines cut by transversal are ||, then the alternate exterior angles are congruent

6

m

n

1

2tnthen

tmnmIf

,

,||

Perpendicular Transversal

If a transversal is perpendicular to one of two

|| lines, then it is perpendicular to the other.

t

1 2

5 6

3 4

7 800

Find the measure of angles 1 – 7 given the information below.

2012 x

y4414 x

Find x, y

1034

52

3243

61

xm

xm

ym

ym

Find x, y, and the measure of all angles

1234

wm

zm

ym

xm

xm

5

204

3

2452

81Find w, x, y, z, and the measure of all angles

1

2 3

4

5

3.4 – Proving Lines are Parallel

Simply stated, the postulates and theorems yesterday have TRUE converses

CAP Conv

|| are lines then the

, are ' corres theIf

Post. Corres of Converse

s

ThrmAIA Conv

|| are lines then the

, are 'int alt theIf

Thrm. Int. Alt. of Converse

s

m

n

p

12

3

p||nthen

31 If p||nthen

32 If

5

ThrmCIA Conv

|| are lines then thesupp, are

'int econsecutiv theIf

Thrm Interior

eConsecutiv of Converse

s

m

n

p

12

3

p||nthen

sup, 4,3 If

4

ThrmAEA Conv

|| are lines the

then congruent, are '

exterior Alternate theIf

Thrm Exterior

Alternate of Converse

s

p||nthen

,51 If

5

m

n

p

1 2

5

3

I show the angles, you say what theorem makes the lines parallel.

4

687

1,5 congruent

3,6 congruent

3,5 supplementary

1,8 congruent

4, 8 congruent

3, 5 congruent

5, 8 congruent

A

Which lines are parallel?

B

C

D

35

35

40

38

l m

You try it! Are l and m parallel? How?

30o

40o

110o

l m

60o

44o

66o

l mn p

40o

80o 50o

80o

discuss

Which lines are parallel? How?

l

m

You try it! What does x have to be for l and m to be parallel?

70o

xo

l

m

(x + 40)o

(3x)o

m

n

p

1 2

5

3 4

687

pn || :Prove

arysupplement

are 8 and 2 :Given

pn || :Prove

81 :Given

CAP) conv

use AEA, Conv uset (Can'

ThrmAEA Conv Proving

3.5 – Using Properties of Parallel Lines

1) Draw a ray

2) Use original vertex, make radius.

3) Transfer radius to the ray you drew, and draw an arc.

4) Set radius from D and E, and transfer it to the new lines, setting the point on F and draw an intersection on the arc, then connect the dots.

Copy an angle.

Given a line and a point, construct a line parallel to the given line through the given point.

1) Pick any point on the line, draw a line from there through the given point.

2) Using the angle formed by the given line and the drawn line, make a congruent angle using the given point as the vertex.

3.6 – Parallel Lines in the Coordinate Plane

x

y

SLOPE = m =

SLOPE = m =y2 – y1

x2 – x1

SLOPE FORMULA!! MEMORIZE!!

Find points and label

Plug into formula

Reduce Fraction

(1, 0) (4, -1)

x1 y1 x2 y2

x

y

SLOPE = m =y2 – y1

x2 – x1

SLOPE FORMULA!! MEMORIZE!!

Find points and label

Plug into formula

Reduce Fraction

(-2, -1) (2, 5)

x1 y1 x2 y2

SLOPE = m =

Postulate: Slopes of Parallel Lines

In a coordinate plane, two nonvertical lines are parallel IFF they have the same slope. Any two vertical lines are parallel.

Basically Same slope means parallel.

Find the slope between each set of points. See which ones match up to be parallel.

(4, 3)

(-2, -1)

(2, 0)

(-1, 3)

(2, 3)

(-2, -1)

(-5, 2)

(-1, -2)

(-1, 3)

(-3, 0)

(1, 2)

(-8, -4)

Slope-intercept form

Point-slope form

bmxy )( 11 xxmyy

Write the equation of the line given a point and a slope in SLOPE-INTERCEPT FORM

2

1m (4,2)

11 yx

Standard form CByAx

formintercept -slopein (8,3) through going and

24xy toparallel line theofequation theWrite

formintercept -slopein 2 ofintercept -y awith

5-x5

4y parallel line theofequation theWrite

bmxy

xxmyy

FormIntercept -Slope

)(Form SlopePoint 11

formintercept -slopein

(-3,1) through going and

2x3

1y toparallel line

theofequation theWrite

)( 11 xxmyy

formintercept -slopein

5-intercept -y with 3-x2y

toparallel

line theofequation theWrite

bmxy

x

y

x

y

formintercept -slopein

(6,-1) through going and

5x3

4y toparallel line

theofequation theWrite

formintercept -slopein (10,-4)

throughgoing and 1x5

2y

toparallel

line theofequation theWrite

)( 11 xxmyy bmxy

Grade of a road, it’s rise over run, then changed into a percent.

2% grade

2

100

3.7 – Perpendicular Lines in the Coordinate Plane

Solve for y, change it to ‘y =‘

)1(32 xy

Distribute

Get y by itself

)62(3

27 xy

Notice how by solving for y, we put it in slope intercept form, now we can find the slope.

Parallel and Perpendicular Lines

Parallel Lines have the ___________ slope

Blue Green What do you notice about the lines and the slope?

Slopes are opposite reciprocals, or slopes multiply to equal -1

Also, vertical and horizontal lines are perpendicular

Parallel Lines, SAME SLOPE

Perpendicular Lines, opposite reciprocal.

State the slopes of the line parallel and perpendicular to the slopes on the left.

Slope Parallel Perpendicular

2

5

2

4

7

Find the slope between each set of points. See which ones match up to be perpendicular.

(4, 3)

(-2, -1)

(2, 0)

(-1, 3)

(2, 3)

(-2, -1)

(-5, 3)

(1, -2)

(-1, 3)

(-3, 0)

(3, 2)

(0, 4)

Find the slope of each line, then pair up the perpendicular and parallel lines.

33

2 xy 632 yx 023 yx 1223 yx

formintercept -slopein (8,3) through going and

24xy toLARPERPENDICU line theofequation theWrite

formintercept -slopein (-2,1) through going and

x3

2y toLARPERPENDICU line theofequation theWrite

formintercept -slopein (-3,1)

throughgoing and 2x3

1y

toLARPERPENDICU line

theofequation theWrite

formintercept -slopein (-4,1)

throughgoing and 3-x2y

toLARPERPENDICU

line theofequation theWrite

x

y

x

y

)( 11 xxmyy bmxy

formintercept -slopein (8,-1)

through going and5x3

4y

toLARPERPENDICU line

theofequation theWrite

formintercept -slopein (10,-4)

throughgoing and 1x5

2y

toLARPERPENDICU

line theofequation theWrite

)( 11 xxmyy bmxy