3 Transversal Lines in Two Parallel Lines.ppt

Parallels
Parallels
 §
§ 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 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 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

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.

A
C
B
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 Transversals
You will learn to identify the relationships among pairs of
interior and exterior angles formed by two parallel lines
and a transversal.

In geometry, a line, line segment, or ray that intersects two or more lines at
different 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
3
4
5
7
6
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
3
4
5
7
6
8
m
l
Parallel Lines
t is a transversal for l and m.
t
1 2
3
4
5
7
6
8
b
c
c
b ||
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
3
4
5
7
6
8
When a transversal intersects two lines, _____ angles are formed.
eight
These angles are given special names.
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.
Consecutive Interior angles are on
the same side of the transversal.
Alternate Exterior angles are
on the opposite sides of the
transversal.

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

1 2
3
4
5
7
6
8
Theorem 4-2
Consecutive
Interior
Angles
consecutive interior angles is _____________.
supplementary
180
5
4 


 180
6
3 




1 2
3
4
5
7
6
8
Theorem 4-3
Alternate
Exterior
Angles
alternate exterior angles is _________.
congruent
7
1 

 8
2 



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

l
m
1 2
3
4
5
7
6
8
t
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
Give three other pairs of corresponding angles that are formed:
4 and 8 3 and 7 2 and 6

Postulate 4-1
Corresponding
Angles
If two parallel lines are cut by a transversal, then each pair
of
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 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
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 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
“_________”.
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.

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 already
know. 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 || b
1
2
a
b

Theorem 4-5
of alternate interior angles is congruent, then the two lines
are _______.
parallel
If 1 2,
then _____
a || b
1
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 || b
1
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 || b
a
b
t

3 Transversal Lines in Two Parallel Lines.ppt

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