G E O M E T R Y
CHAPTER 3
PARALLEL AND
PERPENDICULAR LINES
Notes & Study Guide
CHAPTER 3 NOTES
2
TABLE OF CONTENTS
LINES AND ANGLES........................................................................................... 3
PROOF AND PERPENDICULAR LINES ............................................................. 6
PARALLEL LINES AND TRANSVERSALS ........................................................ 8
PROVING LINES ARE PARALLEL ................................................................... 10
USING PROPERTIES OF PARALLEL LINES ................................................... 13
PARALLEL LINES IN THE COORDINATE PLANE .......................................... 14
PERPENDICULAR LINES IN THE COORDINATE PLANE .............................. 14
EXTRA HOMEWORK EXAMPLES .................................................................... 18
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CHAPTER 3 NOTES
LINES AND ANGLES
3
INTRODUCTION
Notice in Example 1 that, although there are many lines through D that are skew
¯
¯
˘
in˘3Example
1 that,
although
thereDare
lines
thatisto
are
skew
InNotice
Chapter
we examine
parallel
perpendicular
lines.
We
will
how
identify
, there
is only
one
lineand
through
thatmany
is parallel
tothrough
AB learn
andDthere
only
one
to AB
¯
˘
¯
˘
¯
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them
also is
study
their
characteristics
andthat
some
of their unique
features.
Most
of this
,
there
only
one
line
through
D
is
parallel
to
AB
and
there
is
only
one
to ABand
line through D that is perpendicular ¯
to˘AB .
chapter
will focus
on parallel
lines.
line through
D that
is perpendicular
to AB .
HELP
PA R A L L E L A N D P E R P E N D I C U L A R P O S T U L AT E S
LICATION LINK 
PA R A L L E L A N BETWEEN
D PERPEN
D I C U L A R P O S T U L AT E S
RELATIONSHIPS
LINES
POSTULATE 13
Parallel Postulate
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Nt LINK
Any two lines that are coplanar and do not
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POSTULATE
13a line
Parallel
If there is
and a Postulate
point not on
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site
ormation
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arallel
on
are called parallel. P
intersect
the is
line,
thenand
there
is exactly
one
If there
a line
a point
not on
line through
the is
point
parallel
to
the
line,
then
there
exactly
one
Notation:the
small
triangle
given
line. or arrow on the lines
line through the point parallel to
the given line.
POSTULATE 14 Perpendicular Postulate
P
l
l
There is exactly one line
through P parallel to l.
There is exactly one line
through P parallel to l.
Any two lines that intersect to form a 90° angle are
If there is
a line and a point not
on
POSTULATE
14 Perpendicular
Postulate
called
perpendicular.
P
the line, then there is exactly one line
If there
is a line
point not on to
through
the and
pointaperpendicular
the line,
then
there
is
exactly
one line
Notation:
small
square
at
the
intersection
the given line.
through the point perpendicular to
the given line.
l
P
l
There is exactly one line
through P perpendicular to l.
There is exactly one line
Any two lines that do not intersect but are not through P perpendicular to l.
Youare
cancalled
use a compass
coplanar
skew. and a straightedge to construct the line that passes
through a given point and is perpendicular to a given line. In Lesson 6.6, you will
learnuse
why
this construction
works.
You can
a compass
and a straightedge
to construct the line that passes
Notation: none, must write it out
through
given
point
is perpendicular
a given
line. 3.5.
In Lesson 6.6, you will
Youawill
learn
howand
to construct
a paralleltoline
in Lesson
learn why this construction works.
You
will learn
how to construct a parallel line in Lesson 3.5.

POSTUALTES
ACTIVITY
Parallel: Construction
If you have a line and a point (not on that line), then there is only 1 way
A Perpendicular
to atoLine
to drawACTIVITY
a new line through
the point AND parallel
the original line. Use the following steps to construct a line that passes through a given
Construction
Perpendicular:
If you
have
a line and
a point
Perpendicular
tolinea l.Line
point P and
isAperpendicular
to a given
(not on that line), then there is only 1 way to
Useathe
following
steps to construct
draw
new
line through
the pointa line
ANDthat passes through a given
point
P
and
is
perpendicular
to
a
given
line
perpendicular to the
P original line.
P l.
P
A
P
B l
A
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A
B l
Place the compass
point at P and draw
an arc that intersects
twice. Label
Placeline
thelcompass
the intersections A
point at P and draw
and B.
A
1
2
B l
A
P
B l
q
q
B l
Draw
an arc with
q
center A. Using the
same radius, draw an
arc with center B.
Draw
an arc with
Label the intersection
center A. Using the
of the arcs Q.
2
1
P
A
B l
Use qa straightedge
¯
˘
CHAPTER
NOTES
.
to draw3PQ
¯
˘
PQ fi l.
3
3
Use a straightedge
¯
˘
to draw PQ .
4
LINES AND ANGLES
 TRANSVERSALS
Say you have two lines. If a third line intersects those two at the same time (in
different places), then that line is called a transversal. NOTE: The two lines being intersected DO NOT
have to be parallel.
ANGLES FORMED BY A TRANSVERSAL
When a transversal intersects two lines, there are 8
angles that are created. The space in between the two lines being intersected
is referred to as the interior. The other regions are
both exterior.
Within these 8 angles, there are 4 types of special angle pairs. The type of
angle pair depends on where the angles are located.
SPECIAL ANGLE PAIRS
Alternate Interior – two angles in the interior, with
one on each side of the transversal
(angles 3 & 6, angles 4 & 5)
Alternate Exterior – two angles in the exterior, with
one on each side of the transversal
(angles 1 & 8, angles 2 & 7)
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CHAPTER 3 NOTES
LINES AND ANGLES
5
Same-Side Interior – two angles in the interior, with
both on the same side of the transversal
(angles 3 & 5, angles 4 & 6)
Corresponding – two angles (one interior & one
exterior) with both on same side of the transversal
(angles 1 & 5, angles 2 & 6, angles 3 & 7, angles 4 & 8)
 EXAMPLES
Use the figure to identify any angles that fit the description given.
1. Corresponding
2. Alternate Interior
3. Same-side Interior
Given two angles, what type of special angle pair
are they?
1.
2.
3.
4.
Angles 2 and 7
Angles 3 and 6
Angles 4 and 8
Angles 1 and 6
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CHAPTER 3 NOTES
6
PROOF AND PERPENDICULAR LINES
INTRODUCTION
In this section we will look at the 3 main theorems regarding perpendicular lines.
Normally, you would be asked to prove these; however, all we will do is examine what
each one says, recognize when to use it and then ask you to apply them within a
homework-style problem.
 PERPENDICULAR LINE THEOREMS
THEOREM 3.1  If two lines intersect to form a linear
pair of congruent angles, then the lines must be
perpendicular. Linear pairs always total 180°. If the two angles are congruent, then they each must be
90°. Only perpendicular lines can do that.
THEOREM 3.2  If two sides of two adjacent angles are
perpendicular, then the angles must be complementary. When two angles are connected and form an “L”, then the two
angles have to add up to 90°.
THEOREM 3.3  If two lines are perpendicular, then they
intersect to form four right angles. Once you know that 2 lines form one right angle, they automatically
make them all right angles.
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CHAPTER 3 NOTES
PROOF AND PERPENDICULAR LINES
7
 EXAMPLES
For each figure, find the value of the variable.
What can be concluded about the angles?
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CHAPTER 3 NOTES
8
PARALLEL LINES AND TRANSVERSALS
INTRODUCTION
In section 1 we introduced four special angle pairs that exist when a transversal
intersects two other lines. Those lines did not have to be parallel for the angles to exist.
In this section, we will study what happens to the angles when the lines ARE parallel.
 PARALLEL LINES AND TRANSVERSALS
When a transversal intersects two lines, four special
angle pairs are created. If those two lines are parallel then we get extra
information about the angles (look for the marks!)
The angle pair information is summarized as 4 postulates/theorems.
POSTULATE 15 (Corresponding Angles Postulate) If two parallel lines are cut by a transversal, then the
corresponding angles are congruent.
THEOREM 3.4 (Alt. Int. Angle Theorem) If two parallel lines are cut by a transversal, then the
corresponding angles are congruent.
THEOREM 3.5 (Same-side Int. Angle Theorem) If two parallel lines are cut by a transversal, then the
same-side interior angles are supplementary.
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CHAPTER 3 NOTES
DEDUCTIVE REASONING
9
THEOREM 3.6 (Alt. Ext. Angle Theorem) If two parallel lines are cut by a transversal, then the
alternate exterior angles are congruent.
REMEMBER! These four types of angles simply exist whether the lines are
parallel or not.
BUT, you do not know anything about the sizes of those angles unless the lines
are parallel!
EXAMPLES
For each figure, find the value of the variable or the size of the numbered angles.
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CHAPTER 3 NOTES
10
PROVING LINES ARE PARALLEL
INTRODUCTION
In the last section we saw how we could make conclusions about special angle pairs
ONLY IF a transversal was intersecting two PARALLEL lines.
In this section, we will reverse the process. We will learn how to use the sizes of the
special angle pairs to prove that the lines are parallel.
PROVING LINES ARE PARALLEL
When a transversal intersects two parallel lines, we get information about the
angle sizes (all of section 3).
The key is that we needed the parallel lines BEFORE we made the conclusions.
These parallel lines are already
given, so we know the corresponding
angles are congruent.
Same picture. Angles are given but
no parallel lines. How do we know if
the lines really are parallel?
We prove the lines are parallel by using the same four Theorems from the last
section; just in reverse. They are called CONVERSE angle theorems. In other
words…
When we have the parallel lines and want angle sizes, use the regular theorems.
When we have angle sizes and want parallel lines, use the converse theorems.
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CHAPTER 3 NOTES
DEDUCTIVE REASONING
11
THE CONVERSE ANGLE THEOREMS
POSTULATE 16 (Corresponding Angle Converse)
If two lines are cut by a transversal so that the
corresponding angles are congruent, then the lines are
parallel.
THEOREM 3.8 (Alt. Int. Angle Converse)
If two lines are cut by a transversal so that the
alternate interior angles are congruent, then the lines
are parallel.
THEOREM 3.9 (Same-side Int. Angle Converse)
If two lines are cut by a transversal so that the sameside interior angles are supplementary, then the lines
are parallel.
THEOREM 3.10 (Alt. Ext. Angle Converse)
If two lines are cut by a transversal so that the
alternate exterior angles are congruent, then the lines
are parallel.
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CHAPTER 3 NOTES
12
REASONING WITH PROPERTIES FROM ALGEBRA
EXAMPLES
For each figure, is it possible to prove that the lines are parallel? If yes, which
postulate/theorem was used?
For each figure, find the value of x that would make the lines parallel.
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CHAPTER 3 NOTES
USING PROPERTIES OF PARALLEL LINES
13
Here are two important properties of parallel lines. While they are important, it’s
not enough to do assign homework on this section.
All you need to do is know these two properties and be able to recognize them.
 THEOREM 3.11
If two lines are both parallel to the same third line,
then those two lines are parallel to each other. (This is a use of the Transitive Property)
 THEOREM 3.12
In a plane, if two lines are both perpendicular to the
same third line, then those two lines are parallel to
each other. (This is also a use of the Transitive Property)
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CHAPTER 3 NOTES
14
PARALLEL/PERP LINES IN THE COORDINATE PLANE
INTRODUCTION
To end Chapter 3, we will take the same parallel and perpendicular lines we’ve been
working with and put them onto coordinate grids. We did this in Chapter 1 with segments.
We will make use of the same formulas we’ve already used: slope, midpoint & distance.
Our focus here will be with slope.
 SLOPE
From algebra, we know that a number that
represents the steepness of a line is called its
slope. Slope can be found
many ways. There’s
“rise over run” or by
using the actual
coordinates.
TYPES OF SLOPE
POSITIVE: positive number; goes up from left to right
NEGATIVE: negative number: goes down from left to right
ZERO SLOPE: value of zero; horizontal line
NO SLOPE: zero in the denominator; vertical line
EXAMPLES
A(–4, 2)
B(3, –6)
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CHAPTER 3 NOTES
PARALLEL/PERP LINES IN THE COORDINATE PLANE
15
 SLOPES OF PARALLEL LINES
When two lines are parallel, they are both running in the
same direction.
By default, they will have the exact same rise and run.
This leads to an important rule of Geometry…
POSTULATE 17
Two non-vertical lines are parallel if and only if they have the same slopes.
 SLOPES OF PERPENDICULAR LINES
When two lines are perpendicular, they have what are called negative
reciprocal slopes.
Negative means they will have opposite signs (one positive; one negative)
Reciprocal means that one slope will be the flipped version of the other.
–2/3 and 3/2
1/3 and –3/1
–2/7 and 7/2
POSTULATE 18
Two non-vertical lines are perpendicular if and only if they have negative
reciprocal slopes.
To determine parallel or perpendicular, we have to find the slopes of the lines…
 Count rise and run using the grid boxes
 Use the slope formula
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CHAPTER 3 NOTES
16
PARALLEL/PERP LINES IN THE COORDINATE PLANE
EXAMPLES
 EQUATIONS OF LINES
All line on a coordinate plane can be written as algebraic equations. There are
three formats for these equations…
SLOPE-INTERCEPT
POINT-SLOPE
STANDARD
y = mx + b
(y – y1) = m(x – x1)
Ax + By = C
slope = m
slope = m
slope = –A/B
To decide if the lines are parallel or perpendicular from their equations, all you
need to do is to look at their slopes.
TIP! Get the slope from whatever you’re given. Do not try to re-write any
equation unless you absolutely have to!
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CHAPTER 3 NOTES
PARALLEL/PERP LINES IN THE COORDINATE PLANE
17
EXAMPLES
Write the equation of the line from the given information. Determine if the two equations given represent perpendicular lines.
Are the lines parallel, perpendicular or neither?
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CHAPTER 3 NOTES