Parallels

ParallelsParallelsParallelsParallels

§§ 4.1 4.1 Parallel Lines and Planes

§§ 4.4 4.4 Proving Lines Parallel

§§ 4.3 4.3 Transversals and Corresponding Angles

§§ 4.2 4.2 Parallel Lines and Transversals

§§ 4.6 4.6 Equations of Lines

§§ 4.5 4.5 Slope

Parallel Lines and PlanesParallel Lines and Planes

You will learn to describe relationships among lines, parts of lines, and planes.

In geometry, two lines in a plane that are always the same distance apart are ____________.parallel lines

No two parallel lines intersect, no matter how far you extend them.


Definition of

Parallel

Lines

Two lines are parallel iff they are in the same plane and do not ________.intersect


Planes can also be parallel.

The shelves of a bookcase are examples of parts of planes.The shelves are the same distance apart at all points, and do not appear tointersect.

They are _______.parallel

In geometry, planes that do not intersect are called _____________.parallel planes

Q

J

K

M

L

S

R

P

Plane PSR || plane JML

Plane JPQ || plane MLR

Plane PJM || plane QRL


Sometimes lines that do not intersect are not in the same plane.

These lines are called __________.skew lines

Definition of

Skew

Lines

Two lines that are not in the same plane are skew iffthey do not intersect.


A

CB

E

G

H

D

F

Name the parts of the figure:

1) All planes parallel to plane ABF

2) All segments that intersect DH

3) All segments parallel to CD

4) All segments skew to AB

Plane DCG

AD, CD, GH, AH, EH

AB, GH, EF

DH, CG, FG, EH

Parallel Lines and TransversalsParallel Lines and Transversals

You will learn to identify the relationships among pairs of interior and exterior angles formed by two parallel linesand a transversal.


In geometry, a line, line segment, or ray that intersects two or more lines atdifferent points is called a __________transversal

l

m

B

A

AB is an example of a transversal. It intercepts lines l and m.

Note all of the different angles formed at the points of intersection.

1 2

34

5

76

8


Definition of

Transversal

In a plane, a line is a transversal iff it intersects two or more

Lines, each at a different point.

The lines cut by a transversal may or may not be parallel.

l

m

1 2

34

576

8

ml

Parallel Lines

t is a transversal for l and m.

t

1 234

5

7

6

8

b

c

cb ||

Nonparallel Lines

r is a transversal for b and c.

r


Two lines divide the plane into three regions.

The region between the lines is referred to as the interior.

The two regions not between the lines is referred to as the exterior.

Exterior

Exterior

Interior

l

m

1 2

34

576

8


When a transversal intersects two lines, _____ angles are formed.eight

These angles are given special names.

t

Interior angles lie between thetwo lines.

Exterior angles lie outside thetwo lines.

Alternate Interior angles are on the opposite sides of the transversal.

Consectutive Interior angles are on the same side of the transversal.

Alternate Exterior angles areon the opposite sides of thetransversal.


Theorem 4-1

Alternate

Interior

Angles

If two parallel lines are cut by a transversal, then each pair of

Alternate interior angles is _________.

1 234

57

68

64 53

congruent


1 2

34

576

8

Theorem 4-2

Consecutive

Interior

Angles


consecutive interior angles is _____________.supplementary

18054 18063


1 2

34

576

8

Theorem 4-3

Alternate

Exterior

Angles


alternate exterior angles is _________.congruent

71 82

Practice Problems:

1, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, and 46 (total = 23)

Transversals and Corresponding AnglesTransversals and Corresponding Angles

You will learn to identify the relationships among pairs of corresponding angles formed by two parallel lines and atransversal.


l

m

1 2

34

576

8

t

When a transversal crosses two lines, the intersection creates a number ofangles that are related to each other.

Note 1 and 5 below. Although one is an exterior angle and the other is an interior angle, both lie on the same side of the transversal.

Angle 1 and 5 are called __________________.corresponding angles

Give three other pairs of corresponding angles that are formed:

4 and 8 3 and 7 2 and 6


Postulate 4-1

Corresponding

Angles


corresponding angles is _________.congruent


Concept

Summary

Congruent Supplementary

alternate interior

alternate exterior

corresponding

consecutive interior

Types of angle pairs formed when a transversal cuts two parallel lines.


s t

c

d

1 2 3 45 6 7 8

9 10 11 12

13 14 15 16

s || t and c || d.

Name all the angles that arecongruent to 1.Give a reason for each answer.

3 1 corresponding angles

6 1 vertical angles

8 1 alternate exterior angles

9 1 corresponding angles

11 9 1 corresponding angles

14 1 alternate exterior angles

16 14 1 corresponding angles

Practice Problems: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20,

22, 24, 26, 28, 30, 32, 34, 36, and 38 (total = 19)

Proving Lines ParallelProving Lines Parallel

You will learn to identify conditions that produce parallel lines.

Reminder: In Chapter 1, we discussed “if-then” statements (pg. 24).

Within those statements, we identified the “__________” and the “_________”.

hypothesisconclusion

I said then that in mathematics, we only use the term “if and only if”if the converse of the statement is true.


Postulate 4 – 1 (pg. 156):

IF ___________________________________,

THEN ________________________________________.

two parallel lines are cut by a transversal

each pair of corresponding angles is congruent

The postulates used in §4 - 4 are the converse of postulates that you alreadyknow. COOL, HUH?

§4 – 4, Postulate 4 – 2 (pg. 162):

IF ________________________________________,

THEN ____________________________________.

each pair of corresponding angles is congruent

two parallel lines are cut by a transversal


Postulate 4-2

In a plane, if two lines are cut by a transversal so that a pair

of corresponding angles is congruent, then the lines are

_______.parallel

If 1 2,

then _____a || b1

2

a

b


Theorem 4-5


of alternate interior angles is congruent, then the two lines are _______.parallel

If 1 2,

then _____a || b1

2

a

b


Theorem 4-6


of alternate exterior angles is congruent, then the two lines are _______.parallel

If 1 2,

then _____a || b

1

2

a

b


Theorem 4-7


of consecutive interior angles is supplementary, then the two lines are _______.parallel

If 1 + 2 = 180,

then _____a || b1

2

a

b


Theorem 4-8


of consecutive interior angles is supplementary, then the two lines are _______.parallel

If a t and b t,

then _____a || ba

b

t


Concept

Summary

We now have five ways to prove that two lines are parallel.

Show that a pair of corresponding angles is congruent.

Show that a pair of alternate interior angles is congruent.

Show that a pair of alternate exterior angles is congruent.

Show that a pair of consecutive interior angles is supplementary.

Show that two lines in a plane are perpendicular to a third line.


Identify any parallel segments. Explain your reasoning.

G

A

Y

DR

90°

90°

therefore,

GA lar toperpendicuboth are and RDGY

8.-4 Theoremby RDGY


EB

ST

(6x - 26)° (2x + 10)°

(5x + 2)°

Find the value for x so BE || TS.

ES is a transversal for BE and TS.

BES and EST are _________________ angles.consecutive interior

If mBES + mEST = 180, then BE || TS by Theorem 4 – 7.

mBES + mEST = 180

(2x + 10) + (5x + 2) = 180

7x + 12 = 180

7x = 168

x = 24

Thus, if x = 24, then BE || TS.

Practice Problems: 1, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14,

15, 16, 17, 18, 19, 21, 25, and 26 (total = 19)

SlopeSlope

You will learn to find the slopes of lines and use slope to identify parallel and perpendicular lines.

If the pilot doesn’t change something, he / she will not make it home for Christmas. Would you agree?Consider the options:

1) Keep the same slope of his / her path.Not a good choice!

2) Go straight up.

Not possible! This is an airplane, not a helicopter.

There has got to be some “measurable” way to get this aircraftto clear such obstacles.

Discuss how you might radio a pilot and tell him or her how toadjust the slope of their flight path in order to clear the mountain.

Fortunately, there is a way to measure a proper “slope” to clear the obstacle.

We measure the “change in height” requiredand divide that by the “horizontal change” required.

y

x

vertical change

horizontal change

ySlope

x

vertical change 4,000 4 2

horizontal change 10,000 10 5

y ftSlope

x ft

y

x10000

10000

00

SlopeSlope

y

x10

-5 10

10

-5

10

-10

-10

5

5-10

-10

The steepness of a line is called the _____.slope

Slope is defined as the ratio of the ____, or vertical change, to the ___, orhorizontal change, as you move from one point on the line to another.

rise run

SlopeSlope

2 1x x

rise

runm

y

x

The slope m of the non-vertical line passing through the pointsand is

1 1( , )x y2 2( , )x y

1 1( , )x y

2 2( , )x y

2 1y y

ychange in

change x in 2 1

2 1x x

y y

SlopeSlope

Definition

of

Slope

The slope “m” of a line containing two points with coordinates

(x1, y1), and (x2, y2), is given by the formula

scoordinate- xingcorrespond theof difference

scoordinate-y theof difference slope

1212

12 where,x

xxx

yym

SlopeSlope

The slope m of a non-vertical line is the number of units the line rises or fallsfor each unit of horizontal change from left to right.

y

x

(1, 1)

(3, 6)

run = 3 - 1 = 2 units

rise = 6 - 1 = 5 units13

16

m

2

5 m

x)(

)(

run

yrisem

6 & 7

SlopeSlope

Postulate

4 – 3

Two distinct nonvertical lines are parallel iff they have _____________.the same slope

111 bxmy 222 bxmy

21 iffLL

21 mm

SlopeSlope

Postulate

4 – 4

Two nonvertical lines are perpendicular iff ___________________________.the product of their slope is -1

111 bxmy 222 bxmy

21 iffLL 121 mm

8 & 9

Practice Problems: 1, 3, 4, 5, 6, 7, 8, 9, 10, 12,

14, 16, 17, 20, 22, 24, 26, 30, and 32 (total = 19)

Equations of LinesEquations of Lines

You will learn to write and graph equations of lines.

The equation y = 2x – 1 is called a _____________ because its graph is a straight line.

linear equation

We can substitute different values for x in the graph to find correspondingvalues for y.

0

y

0 x

81 3 5 7-1-1

2

4

6

8

-1 4 8

1

5

-1 6

3

2

7

8

x y = 2x -1 y

1

2

3

y = 2(1) -1 1

3

5

y = 2(2) -1

y = 2(3) -1 (1, 1)

(2, 3)

(3, 5)There are many more points whose orderedpairs are solutions of y = 2x – 1. These points also lie on the line.


0

y

0 x

5-2 1 3 5

5

-2

1

3

5

-3 2-3

-1

4

-1-3

-3

2

4

y = 2x – 1

Look at the graph of y = 2x – 1 .

The y – value of the point where the line crosses the y-axis is ___.- 1

This value is called the ____________ of the line.y - intercept

(0, -1)

Most linear equations can be written in the form __________.y = mx + b

This form is called the ___________________.slope – intercept form

y = mx + bslope y - intercept


Slope – Intercept

Form

An equation of the line having slope m and y-intercept b is

y = mx + b


1) Rewrite the equation in slope – intercept form by solving for y.

2x – 3 y = 18

2) Graph 2x + y = 3 using the slope and y – intercept.

y = –2x + 3

0

y

0 x

5-2 1 3 5

5

-2

1

3

5

-3 2-3

-1

4

-1-3

-3

2

4

1) Identify and graph the y-intercept.

2) Follow the slope a second point on the line.

(0, 3)

(1, 1)

3) Draw the line between the two points.


1) Write an equation of the line parallel to the graph of y = 2x – 5 that passes through the point (3, 7).

2) Write an equation of the line parallel to the graph of 3x + y = 6 that passes through the point (1, 4).

3) Write an equation of the line perpendicualr to the graph of that passes through the point ( - 3, 8).

54

1 xy

y = 2x + 1

y = -3x + 7

y = -4x -4

Practice Problems:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 40, and 42 (total = 24)

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