1. 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

2. 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.

3. Parallel Lines and Planes
Definition of
Parallel
Lines
Two lines are parallel iff they are in the same plane and do not ________.
intersect

4. Parallel Lines and Planes
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.
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

5. Parallel Lines and Planes
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 iff they do not intersect.

6. Parallel Lines and Planes
Name the parts of the figure:
1) All planes parallel to plane ABF
Plane DCG
2) All segments that intersect A C B E G H D F
AD, CD, GH, AH, EH
3) All segments parallel to
AB, GH, EF
4) All segments skew to
DH, CG, FG, EH

7. End of Section 4.1

8. 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 lines and a transversal.

9. 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 B A l m 1 2 4 3 5 6 8 7
is an example of a transversal. It intercepts lines l and m.
Note all of the different angles formed at the points of intersection.

10. Parallel Lines and Transversals
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 3 4 5 7 6 8
Parallel Lines
t is a transversal for l and m. t 1 2 3 4 5 7 6 8 b c
Nonparallel Lines
r is a transversal for b and c. r

11. 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.
Exterior
Interior

12. Parallel Lines and Transversals
When a transversal intersects two lines, _____ angles are formed.
eight
These angles are given special names. l m 1 2 3 4 5 7 6 8 t
Interior angles lie between the
two lines.
Exterior angles lie outside 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. ?

13. Parallel Lines and Transversals
Theorem 4-1
Alternate
Interior
Angles
If two parallel lines are cut by a transversal, then each pair of
Alternate interior angles is _________.
congruent 1 2 4 3 5 6 8 7

14. Parallel Lines and Transversals
Theorem 4-2
Consecutive
Interior
Angles
If two parallel lines are cut by a transversal, then each pair of
consecutive interior angles is _____________.
supplementary 1 2 3 4 5 7 6 8

15. Parallel Lines and Transversals
Theorem 4-3
Alternate
Exterior
Angles
If two parallel lines are cut by a transversal, then each pair of
alternate exterior angles is _________.
congruent 1 2 3 4 5 7 6 8 ?

16. 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 (total = 23)

17. 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.

18. Transversals and Corresponding Angles
When a transversal crosses two lines, the intersection creates a number of angles 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 l m 1 2 3 4 5 7 6 8 t
Give three other pairs of corresponding angles that are formed:
4 and 8
3 and 7
2 and 6

19. Transversals and Corresponding Angles
Postulate 4-1
Corresponding
Angles
If two parallel lines are cut by a transversal, then each pair of
corresponding angles is _________.
congruent

20. Transversals and Corresponding Angles
Concept
Summary
Congruent
Supplementary
Types of angle pairs formed when
a transversal cuts two parallel lines.
alternate interior
consecutive interior
alternate exterior
corresponding

21. Transversals and Corresponding Angles1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
s || t and c || d.
Name all the angles that are congruent 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
14  1
alternate exterior angles
11  9  1
corresponding angles
16  14  1
corresponding angles

22. 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)

23. 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).
Within those statements, we identified the “__________” and the
“_________”.
hypothesis
conclusion
I said then that in mathematics, we only use the term
“if and only if”
if the converse of the statement is true.

24. 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 ____________________________________.

25. Proving Lines Parallel
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 1 2 a b
If 1 2,
then _____
a || b

26. Proving Lines Parallel
Theorem 4-5
In a plane, if two lines are cut by a transversal so that a pair
of alternate interior angles is congruent, then the two lines are _______.
parallel 1 2 a b
If 1 2,
then _____
a || b

27. Proving Lines Parallel
Theorem 4-6
In a plane, if two lines are cut by a transversal so that a pair
of alternate exterior angles is congruent, then the two lines are _______.
parallel 1 2 a b
If 1 2,
then _____
a || b

28. Proving Lines Parallel
Theorem 4-7
In a plane, if two lines are cut by a transversal so that a pair
of consecutive interior angles is supplementary, then the two lines are _______.
parallel 1 2 a b
If 1 + 2 = 180,
then _____
a || b

29. Proving Lines Parallel
Theorem 4-8
In a plane, if two lines are cut by a transversal so that a pair
of consecutive interior angles is supplementary, then the two lines are _______.
parallel
If a  t and b  t,
then _____ a b t
a || b

30. Proving Lines Parallel
We now have five ways to prove that two lines are parallel.
Concept
Summary
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.

31. Proving Lines Parallel
Identify any parallel segments. Explain your reasoning. G A Y D R
90°

32. Proving Lines Parallel
Find the value for x so BE || TS. E B S T
(6x - 26)°
(2x + 10)°
(5x + 2)°
ES is a transversal for BE and TS.
mBES + mEST = 180
BES and EST are _________________ angles.
(2x + 10) + (5x + 2) = 180
consecutive interior
7x = 180
If mBES + mEST = 180, then BE || TS by Theorem 4 – 7.
7x = 168
x = 24
Thus, if x = 24, then BE || TS.

33. 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)

34. Slope
What You'll Learn
You will learn to find the slopes of lines and use slope to
identify parallel and perpendicular lines.

35. There has got to be some “measurable” way to get this aircraft to clear such obstacles.
Discuss how you might radio a pilot and tell him or her how to adjust 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.

36. Fortunately, there is a way to measure a proper “slope” to clear the obstacle.
We measure the “change in height” required and divide that by the “horizontal change” required.

37. y x
10000

38. The steepness of a line is called the _____. slope
Slope is defined as the ratio of the ____, or vertical change, to the ___, or horizontal change, as you move from one point on the line to another.
rise
run y x 10 -5
-10 5

39. The slope m of the non-vertical line passing through the points and isy x

40. The slope “m” of a line containing two points with coordinates
Definition of
Slope
The slope “m” of a line containing two points with coordinates
(x1, y1), and (x2, y2), is given by the formula

41. Slope
The slope m of a non-vertical line is the number of units the line rises or falls for each unit of horizontal change from left to right. y x
(3, 6)
rise = = 5 units
(1, 1)
run = = 2 units ?
6 & 7

42. Slope
Postulate
4 – 3
Two distinct nonvertical lines are parallel iff they have _____________.
the same slope

43. Slope
Postulate
4 – 4
Two nonvertical lines are perpendicular iff ___________________________.
the product of their slope is -1 ?
8 & 9

44. 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)

45. Equations of Lines
What You'll Learn
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 corresponding
values for y. y x 8 1 3 5 7 -1 2 4 6 x
y = 2x -1 y
There are many more points whose ordered
pairs are solutions of y = 2x – 1. These points also lie on the line.
(3, 5) 1
y = 2(1) -1 1 2
(2, 3)
y = 2(2) -1 3 3
y = 2(3) -1 5
(1, 1)

46. y = mx + b Look at the graph of y = 2x – 1 .
Equations of Lines
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
Most linear equations can be written in the form __________.
y = mx + b
This form is called the ___________________.
slope – intercept form y x 5 -2 1 3 -3 2 -1 4
y = 2x – 1
y = mx + b
slope
y - intercept
(0, -1)

47. An equation of the line having slope m and y-intercept b is y = mx + b
Equations of Lines
Slope – Intercept
Form
An equation of the line having slope m and y-intercept b is
y = mx + b

48. 1) Rewrite the equation in slope – intercept form by solving for y.
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 y x 5 -2 1 3 -3 2 -1 4
1) Identify and graph the y-intercept.
(0, 3)
2) Follow the slope a second point on the line.
(1, 1)
3) Draw the line between the two points.

49. 3) Write an equation of the line perpendicualr to the graph of
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
3) Write an equation of the line perpendicualr to the graph of
that passes through the point ( - 3, 8).
y = -4x -4

50. 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 (total = 24)