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Parallels

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Parallels

- § 4.1 Parallel Lines and Planes

- § 4.2 Parallel Lines and Transversals

- § 4.3 Transversals and Corresponding Angles

- § 4.4 Proving Lines Parallel

- § 4.5 Slope

- § 4.6 Equations of Lines

Parallel Lines and Planes

What You'll Learn

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.

Parallel Lines and Planes

intersect

Q

R

Parallel Lines and Planes

S

P

K

L

M

J

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 to

intersect.

parallel

They are _______.

parallel planes

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

Plane PSR || plane JML

Plane JPQ || plane MLR

Plane PJM || plane QRL

Parallel Lines and Planes

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

These lines are called __________.

skew lines

2) All segments that intersect

B

C

A

D

3) All segments parallel to

4) All segments skew to

Parallel Lines and Planes

F

G

DH, CG, FG, EH

AB, GH, EF

AD, CD, GH, AH, EH

E

H

Name the parts of the figure:

1) All planes parallel to plane ABF

Plane DCG

End of Section 4.1

Parallel Lines and Transversals

What You'll Learn

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

A

l

m

B

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

Parallel Lines and Transversals

In geometry, a line, line segment, or ray that intersects two or more lines at

different points is called a __________

transversal

2

1

4

3

5

6

8

7

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

Parallel Lines

Nonparallel Lines

b

l

2

1

2

1

3

4

4

3

m

c

6

5

6

5

8

7

7

8

t

r

t is a transversal for l and m.

r is a transversal for b and c.

Parallel Lines and Transversals

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

Exterior

Interior

Exterior

Parallel Lines and Transversals

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.

l

2

1

4

3

m

6

5

8

7

Parallel Lines and Transversals

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

eight

These angles are given special names.

t

Exterior angles lie outside the

two lines.

Interior angles lie between the

two lines.

Alternate Interior angles are on the

opposite sides of the transversal.

Alternate Exterior angles are

on the opposite sides of the

transversal.

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

?

Parallel Lines and Transversals

congruent

2

1

4

3

6

5

7

8

2

1

4

3

6

5

8

7

Parallel Lines and Transversals

supplementary

2

1

4

3

6

5

8

7

Parallel Lines and Transversals

congruent

?

End of Lesson

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 Angles

What You'll Learn

You will learn to identify the relationships among pairs of

corresponding angles formed by two parallel lines and a

transversal.

l

2

1

4

3

m

6

5

8

7

t

Transversals and Corresponding Angles

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.

corresponding angles

Angle 1 and 5 are called __________________.

Give three other pairs of corresponding angles that are formed:

4 and 8

3 and 7

2 and 6

Transversals and Corresponding Angles

congruent

Transversals and Corresponding Angles

Types of angle pairs formed when

a transversal cuts two parallel lines.

consecutive interior

alternate interior

alternate exterior

corresponding

s

t

c

1

3

4

2

5

6

7

8

9

12

11

d

10

14

13

15

16

Transversals and Corresponding Angles

s || t and c || d.

Name all the angles that arecongruent to 1.

Give a reason for each answer.

corresponding angles

3 1

vertical angles

6 1

alternate exterior angles

8 1

corresponding angles

9 1

alternate exterior angles

14 1

corresponding angles

11 9 1

corresponding angles

16 14 1

End of Lesson

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 Parallel

What You'll Learn

You will learn to identify conditions that produce parallel lines.

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

hypothesis

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

“_________”.

conclusion

I said then that in mathematics, we only use the term

“if and only if”

if the converse of the statement is true.

Proving Lines Parallel

Postulate 4 – 1 (pg. 156):

IF ___________________________________,

THEN ________________________________________.

two parallel lines are cut by a transversal

two parallel lines are cut by a transversal

each pair of corresponding angles is congruent

each pair of corresponding angles is congruent

The postulates used in §4 - 4 are the converse of postulates that you already

know.

COOL, HUH?

§4 – 4, Postulate 4 – 2 (pg. 162):

IF ________________________________________,

THEN ____________________________________.

1

a

2

b

Proving Lines Parallel

parallel

If 1 2,

then _____

a || b

a

1

2

b

Proving Lines Parallel

parallel

If 1 2,

then _____

a || b

1

a

b

2

Proving Lines Parallel

parallel

If 1 2,

then _____

a || b

a

1

2

b

Proving Lines Parallel

parallel

If 1 + 2 = 180,

then _____

a || b

t

a

b

Proving Lines Parallel

parallel

If a t andb t,

then _____

a || b

Proving Lines Parallel

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.

Y

G

90°

R

D

90°

A

Proving Lines Parallel

Identify any parallel segments. Explain your reasoning.

Find the value for x so BE || TS.

E

B

(2x + 10)°

(6x - 26)°

(5x + 2)°

T

S

ES is a transversal for BE and TS.

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

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

Proving Lines Parallel

mBES + mEST = 180

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

consecutive interior

BES and EST are _________________ angles.

7x + 12 = 180

7x = 168

x = 24

End of Lesson

Practice Problems:

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

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

Slope

What You'll Learn

You will learn to find the slopes of lines and use slope to

identify parallel and perpendicular lines.

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.

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.

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

10000

x

0

0

10000

y

10

10

5

x

-10

-10

-5

5

10

10

-5

-10

-10

Slope

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.

run

rise

y

x

Slope

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

Slope

y

(3, 6)

(1, 1)

?

6 & 7

x

Slope

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

rise = 6 - 1 = 5 units

run = 3 - 1 = 2 units

Slope

the same slope

?

8 & 9

Slope

the product of their slope is -1

End of Lesson

Practice Problems:

1, 3, 4, 5, 6, 7, 8, 9, 10, 12,

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

y

8

8

7

6

5

4

(3, 5)

(2, 3)

3

(1, 1)

2

1

x

0

-1

-1

-1

-1

1

2

3

4

5

6

7

8

8

0

Equations of Lines

What You'll Learn

You will learn to write and graph equations of lines.

linear equation

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

a straight line.

We can substitute different values for x in the graph to find corresponding

values for y.

There are many more points whose ordered

pairs are solutions of y = 2x – 1. These points also lie on the line.

1

1

y = 2(1) -1

2

3

y = 2(2) -1

3

5

y = 2(3) -1

y = 2x – 1

y

5

5

y - intercept

slope

4

(0, -1)

3

2

1

x

0

-1

-2

-3

-3

-3

-3

-2

-1

1

2

3

4

5

5

0

Equations of Lines

Look at the graph of y = 2x – 1 .

- 1

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

y - intercept

This value is called the ____________ of the line.

y = mx + b

Most linear equations can be written in the form __________.

slope – intercept form

This form is called the ___________________.

y = mx + b

Equations of Lines

y

5

5

4

(0, 3)

(1, 1)

3

2

1

x

0

-1

-2

-3

-3

-3

-3

-2

-1

1

2

3

4

5

5

0

Equations of Lines

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

1) Identify and graph the y-intercept.

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

3) Draw the line between the two points.

3) Write an equation of the line perpendicualr to the graph of

that passes through the point ( - 3, 8).

Equations of Lines

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

y = 2x + 1

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

y = -3x + 7

y = -4x -4

End of Lesson

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)