Download Chapter 7.1 (Parallel Lines and Transversals)

DRILL
Name the parts of the figure:
1) All planes parallel to plane ABF
Plane DCG
B
2) All segments that intersect DH
AD, CD, GH, AH, EH
3) All segments parallel to
C
A
D
CD
AB, GH, EF
4) All segments skew to AB
DH, CG, FG, EH
F
E
G
H
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 Transversals
You will learn to identify the relationships among pairs of
interior and exterior angles formed by two parallel lines
and a transversal.
Parallel Lines and Transversals
In geometry, a line, line segment, or ray that intersects two or more lines at
transversal
different points is called a __________
A
2
1
4
5
8
6
7
3
l
m
B
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.
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.
Parallel Lines
Nonparallel Lines
l
1 2
4 3
lm
t
1 2
4 3
m
5 6
8 7
c
5 6
8 7
b || c
t
is a transversal for l and m.
b
r
r
is a transversal for b and c.
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
Exterior
Parallel Lines and Transversals
eight angles are formed.
When a transversal intersects two lines, _____
These angles are given special names.
l
1 2
4 3
m
5 6
8 7
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.
Parallel Lines and Transversals
Theorem:
Alternate
Interior
Angles
If two parallel lines are cut by a transversal, then each pair of
congruent
Alternate interior angles is _________.
1 2
4 3
5 6
8 7
4  6
3  5
Parallel Lines and Transversals
Theorem:
If two parallel lines are cut by a transversal, then each pair of
supplementary
Consecutive consecutive interior angles is _____________.
Interior
Angles
1 2
4 3
5 6
8 7
4  5  180
3  6  180
Parallel Lines and Transversals
Theorem:
Alternate
Exterior
Angles
If two parallel lines are cut by a transversal, then each pair of
congruent
alternate exterior angles is _________.
1 2
4 3
5 6
8 7
1  7
2  8
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.
corresponding angles
Angle 1 and 5 are called __________________.
l
1 2
4 3
m
5 6
8 7
t
Give three other pairs of corresponding angles that are formed:
4 and 8
3 and 7
2 and 6
Transversals and Corresponding Angles
Postulate:
If two parallel lines are cut by a transversal, then each pair of
congruent
Corresponding corresponding angles is _________.
Angles
Transversals and Corresponding Angles
Types of angle pairs formed when
a transversal cuts two parallel lines.
Concept
Summary
Congruent
Supplementary
alternate interior
consecutive interior
alternate exterior
corresponding
5
Transversals and Corresponding Angles
s
s || t and c || d.
Name all the angles that are
congruent to 1.
Give a reason for each answer.
1 2
5 6
9
10
13 14
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
t
3
7
11 12
c
4
8
d
15 16
6 – 16
Proving Lines Parallel
Postulate 7 – 1 (pg. 364):
two parallel lines are cut by a transversal
IF ___________________________________,
each pair of corresponding angles is congruent
THEN ________________________________________.
Converse of that statement (Tomorrow)
IF ________________________________________,
THEN ____________________________________.
Proving Lines Parallel
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
_______.
Postulate 7-1
1
2
If 1 2,
a
b
a || b
then _____
Proving Lines Parallel
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
parallel
are _______.
Theorem 7-1
If 1 2,
a
2
1
b
a || b
then _____
Proving Lines Parallel
In a plane, if two lines are cut by a transversal so that a pair
of consecutive interior angles is supplementary, then the two
parallel
lines are _______.
Theorem 7-2
If 1 + 2 = 180,
1
2
a
b
a || b
then _____
Proving Lines Parallel
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
parallel
are _______.
Theorem 7-3
1
2
If 1 2,
a
b
a || b
then _____
Proving Lines Parallel
In a plane, if two lines are perpendicular to a third line, then
parallel
the two lines are _______.
Theorem 4-8
If a  t and b  t,
t
a
b
a || b
then _____
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.
Concept
Summary 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.