Download Parallel Lines and Transversals

Parallels

§ 4.2 Parallel Lines and Transversals

§ 4.3 Transversals and Corresponding Angles
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§ 4.4 Proving Lines Parallel
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
parallel lines
distance apart are ____________.
No two parallel lines intersect, no matter how far you extend them.
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.
Same Side Interior angles are on
the same side of the transversal.
Parallel Lines and Transversals
Theorem 3-1 If two parallel lines are cut by a transversal, then each pair of
congruent
Alternate
Alternate interior angles is _________.
Interior
Angles
1 2
4 3
5 6
8 7
4  6
3  5
Parallel Lines and Transversals
Theorem 3-2 If two parallel lines are cut by a transversal, then each pair of
supplementary
Same Side Same side interior angles is _____________.
Interior
Angles
1 2
4 3
5 6
8 7
4  5  180
3  6  180
Parallel Lines and Transversals
Theorem 3-3 If two parallel lines are cut by a transversal, then each pair of
congruent
Alternate
alternate exterior angles is _________.
Exterior
Angles
1 2
4 3
5 6
8 7
1  7
2  8
Transversals and Corresponding Angles
You will learn to identify the relationships among pairs of
corresponding angles formed by two parallel lines and a
transversal.
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 3-1 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
alternate interior
alternate exterior
corresponding
Supplementary
Same side interior
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
15 16
c
4
8
d
Proving Lines Parallel
You will learn to identify conditions that produce parallel lines.
Reminder: In Chapter 1, we discussed “if-then” statements (pg. 24).
hypothesis and the
Within those statements, we identified 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 3 – 1 (pg. 116):
two parallel lines are cut by a transversal
IF ___________________________________,
each pair of corresponding angles is congruent
THEN ________________________________________.
The postulates used are the converse of postulates that you already
know. COOL, HUH?
, Postulate 3 – 2 (pg. 122):
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 3-2
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 3-3
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 alternate exterior angles is congruent, then the two lines
parallel
are _______.
Theorem 3-6
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 same side interior angles is supplementary, then the two
parallel
lines are _______.
Theorem 3-4
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 same side interior angles is supplementary, then the two
parallel
lines are _______.
Theorem 3-5
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 same side interior angles is
supplementary.
Show that two lines in a plane are perpendicular to a
third line.
Proving Lines Parallel
Identify any parallel segments. Explain your reasoning.
GY and RD are both perpendicu lar to GA
therefore, GY RD by Theorem 4 - 8.
G
R
A
Y
90°
90°
D
Proving Lines Parallel
B
Find the value for x so BE || TS.
T
(6x - 26)°
(2x + 10)°
(5x + 2)°
ES is
a transversal
for BE= and
mBES
+ mEST
180TS.
(2x + and
10) EST
+ (5x are
+ 2)_________________
= 180 side interior angles.
(same
BES
7x + 12 = 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.
S
E