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Geometry Unit 3 – Notes
Parallel and Perpendicular Lines
Review Concepts: Equations of Lines
Syllabus Objective: 3.1 - The student will differentiate among parallel, perpendicular, and
skew lines.
Lines that DO NOT intersect:
o Parallel Lines - coplanar lines that do not intersect. (symbol:║)
• Indicate correct diagram markings for parallel lines.
•
•
B
A
•
C
•D


AB and CD are parallel lines.
o Skew Lines - non-coplanar lines that do not intersect.
•E
•
A
•
B
•
C
•D


AE and CD are skew lines.
Lines that DO intersect:
o Perpendicular Lines – coplanar lines that intersect to form a right angle.
(symbol: ⊥ )
• Indicate correct diagram markings for perpendicular lines.
Remind students that intersecting lines are always coplanar.
Two lines can be related in 3 different ways: intersecting, parallel
or skew. Intersecting lines may or may not be perpendicular.
Two planes can only be related in 2 ways: parallel or intersecting.
Intersecting and parallel lines are coplanar, by definition.
Unit 3
Parallel and Perpendicular Lines (Revised 2010)
Page 1 of 22
Parallel Postulate: If there is a line and a point not on the line, then there is exactly one line
through the point parallel to the given line.
Perpendicular Postulate: If there is a line and a point not on the line, then there is exactly one
line through the point perpendicular to the given line.
Syllabus Objective: 3.2 - The student will analyze relationships when two lines are cut by
a transversal.
Angle Pair Relationships
Transversal - a line that intersects two or more coplanar lines at different points.
 Angle pairs created by a transversal.
• Corresponding - angles in the same relative position, same side of the transversal
and on the same side of the given lines.
• Alternate - opposite sides of the transversal.
• Consecutive - same side of the transversal. (also called same-side)
• Interior - between two lines cut by a transversal.
• Exterior - outside of two lines cut by a transversal.
In the figure to the left,
In the
figures following, all lines are transversals.
l
1
t is a transversal.
2
4
3
6
5
7
8
Angle pairs can be defined by linking two terms
together, such as: Alternate Interior Angles – a
pair of angles on opposite sides of the transversal
and between the two intersected lines.
When identifying these angle pairs, it is helpful to
identify the transversal first.
Ex: ∠1 and ∠5 are corresponding angles,
∠3 and ∠6 are alternate interior angles,
∠2 and ∠8 are same-side exterior angles.
Unit 3
Parallel and Perpendicular Lines (Revised 2010)
Page 2 of 22
1 2
3 4
10
9
11
17
12
18
19
20
26
25
14
13
5 6
7
15
16
21
23
27
28
22
24
8
Examples:
1. Name two same-side interior angles to ∠4 and their transversals. {11, s and 6,
m}
2. Name two corresponding angles to ∠12 and their transversals. {4, s and 16, n}
3. Name the alternate exterior angle to ∠14 and its transversal. {7, t }
4. Name two same-side interior angles to ∠20 and their transversals. {25, w and
22, u}
5. Name two corresponding angles to ∠23 and their transversals. {19, u and 27, v}
6. Name two alternate exterior angles to ∠28 and their transversals. {21, v and
19, w}
7. Name the type of angle pair of the following angles:
a. ∠11 and ∠14 {alternate interior angles}
b. ∠1 and ∠5 {corresponding angles}
c. ∠6 and ∠7 {vertical angles}
d. ∠6 and ∠10 {no relationship}
e. ∠20 and ∠27 {corresponding angles}
f. ∠24 and ∠27 {same-side interior angles}
g. ∠19 and ∠20 {linear pair or supplementary angles}
Unit 3
Parallel and Perpendicular Lines (Revised 2010)
Page 3 of 22
Special Angle Pairs (Postulate and Theorems)
o Corresponding Angles Postulate: If two parallel lines are cut by a transversal, then the
corresponding angles are congruent. {CA Post.}
o Alternate Interior Angles Theorem: If two parallel lines are cut by a transversal, then
the pairs of alternate interior angles are congruent. {AIA Th.}
o Consecutive Interior Angles Theorem: If two parallel lines are cut by a transversal,
then the pairs of consecutive interior angles are supplementary. {CIA Th.}
o Alternate Exterior Angles Theorem: If two parallel lines are cut by a transversal, then
the pairs of alternate exterior angles are congruent. {AEA Th.}
The theorems all follow directly from the postulate.
Consecutive Interior Angles can also be called “Same Side” Interior Angles.
However, “consecutive exterior angles” would be a misnomer as none of the
exterior angles are actually consecutive.
It is a simple exercise to write a proof of “Same Side Exterior Angles.”
This provides an opportunity for students to develop their own useful
theorem.
In simple terms, any two angles formed by the transversal and two parallel
lines can have only one of two relationships: a) they are congruent, or b)
they are supplementary.
Semester Exam Review (Practice Test 08-09)
In the figure below, line m is a transversal.
1
Which best describes the pair of
angles: ∠1 and ∠ 2 ?
A. alternate exterior
B. alternate interior
2
m
Unit 3
C. corresponding
D. vertical
Parallel and Perpendicular Lines (Revised 2010)
Page 4 of 22
Syllabus Objective: 3.3 – The student will solve problems which involve parallel or
perpendicular lines using algebraic techniques.
Postulates:
 Parallel Postulate: If there is a line and a point not on the line, then there is exactly one
line through the point parallel to the given line.
Perpendicular Postulate: If there is a line and a point not on the line, then there is
exactly one line through the point perpendicular to the given line.

Examples: Given the information, find the measure of each numbered angle.
Which theorem or postulate did you use to calculate your solutions?
1. m ∠1= 113°
{ m ∠1, 4, 5, 8 =°
113 and m ∠2,3, 6, 7 =°
67 }
mn 1 2
2. m ∠7 = 47°
4
3
{ m ∠1, 4, 5, 8 =
133° and m ∠2,3, 6, 7 =°
47 }
3. m ∠8
= (2x + 1)°, m∠4
= (3x-61)°
2x + 1= 3x − 61 , x = 62
{ m ∠1, 4, 5, 8 =
125° and m ∠2,3, 6, 7 =°
55 }
4. m ∠5
= (2x + 1)°, m∠2
= (3x-61)°
2x + 1 + 3x − 61 =
180 , x = 48
{ m ∠1, 4, 5, 8 =°
97 and m ∠2,3, 6, 7 =°
83 }
m
5 6
8
7
n
t
Semester Exam Review (Practice Test 08-09)
In the figure below, n  m and l is a transversal.
64°
n
( 4 x − 16 ) °
m
l
What is the value of x?
A. 33
B. 29
C. 20
D. 16
Unit 3
Parallel and Perpendicular Lines (Revised 2010)
Page 5 of 22
Semester Exam Review (Practice Test 08-09)
In the figure below, n  m and l is a transversal.
m
x°
n
117
°
l
What is the value of x?
A. 180
B. 117
C. 63
D. 53
Semester Exam Review (Practice Test 08-09)
In the figure below, m∠FGH =°
65 .
( 2 x − 17 ) °
l
m
G
F
65°
H
What value of x would make line l parallel to line m?
A. 41
B. 49
C. 65
D. 66
Unit 3
Parallel and Perpendicular Lines (Revised 2010)
Page 6 of 22
Semester Exam Review (Practice Test 08-09)
In the figure below, lines l and m are parallel.
5
1
6
2
l
3
4
m
7
8
Which statement is true?
A. ∠1 and ∠3 are congruent.
B. ∠1 and ∠8 are supplementary.
C. ∠2 and ∠4 are supplementary.
D. ∠6 and ∠7 are congruent.
Sample Nevada High School Proficiency Exam question(s): Taken from the 2009
Instructional Materials for the NHSPE provided by the Nevada Department of
Education.



In the diagram below, PF and MH are parallel lines intersected by EK .
What is the measure of ∠GJH ?
A
B
C
D
Unit 3
368
486
730
732
Parallel and Perpendicular Lines (Revised 2010)
Page 7 of 22
Sample Nevada High School Proficiency Exam question(s): Taken from the 2009
Instructional Materials for the NHSPE provided by the Nevada Department of
Education.
In the figure below, TV || ZW , UY || VX , UX || VW , and TZ || VW .
What is the measure of ∠ZYU ?
A 80°
B 100°
C 120°
D 140°
Sample SAT Question(s): Taken from College Board online practice problems.
1. In the figure above, l || m . Which of the following must equal 180?
(A) k + n + r
(B) k + p + s
(C) n + p + s
(D) n + p + t
(E) r + s + t
Unit 3
Parallel and Perpendicular Lines (Revised 2010)
Page 8 of 22
2. Three parallel lines in a plane are intersected by a fourth line, forming twelve angles. If one of the
angles has measure 28°, how many of the other eleven angles have measure 28°?
Grid-In
Syllabus Objective: 3.4 – The student will write proofs relating to parallel and
perpendicular lines.
Theorems of Perpendicular Lines
o If two lines are perpendicular, then they intersect to form four right angles.
This follows directly from the definition of perpendicular lines, after
applying the linear pair postulate and the vertical angle theorem.
o Perpendicular Transversal Theorem – If a transversal is perpendicular to one of two
parallel lines, then it is perpendicular to the other. {P.T. Th.}
Unit 3
Parallel and Perpendicular Lines (Revised 2010)
Page 9 of 22
Given:
mn
and
mn
m⊥t
m
Prove: n ⊥ t
3
(Note: diagram is NOT drawn to scale)
7
1) m  n
1) Given
2)
3)
4)
5)
6)
7)
2)
3)
4)
5)
6)
7)
8) ∠5 is supplementary to ∠3
9) m ⊥ t
10) m ∠1= 90°
11) m ∠5 = 90°
12) n ⊥ t {proof of: P.T. Th.}
Unit 3
{proof of: CIA Th.}
2
4
5 6
n
∠1 ≅ ∠5
∠1 ≅ ∠4
∠5 ≅ ∠4 {proof of: AIA Th.}
∠8 ≅ ∠4
∠1 ≅ ∠8 {proof of: AEA Th.}
∠1 is supplementary to ∠3
1
8
CA. Post.
V. A. Th.
Substitution
CA.s Post.
Substitution
L. P. Post.
8) Substitution
9) Given
10) Def. of ⊥ lines
11) Substitution
12) Def. of ⊥ lines
Parallel and Perpendicular Lines (Revised 2010)
Page 10 of 22
Additional Theorems About Perpendicular Lines
o Theorems:
 If two lines intersect to form a linear pair of congruent, angles, then the lines are
perpendicular.
A•
We are given that the two intersecting


lines, AB and CD, form congruent adjacent angles.
•D
Therefore ∠ACD ≅ ∠DCB .
So by the definition of congruent,
m ∠ACD =
m ∠DCB .
C
m ∠ACB =180° by definition of a straight angle.
B
•
The Angle Addition Postulate says that
m ∠ACD + m ∠DCB = m ∠ACB hence,
m ∠ACD + m ∠DCB = 180°.
Using substitution we obtain
m ∠ACD + m ∠ACD = 180° , which simplifies to
2m ∠ACD =180°.
And the division property of equality allows us to
conclude that m ∠ACD =
90°.

If two sides of two adjacent acute angles are perpendicular, then the angles are
complementary.
 
We are given AB ⊥ BC , hence, m ∠ABC =
90°.
D•
A•
•C
B
Unit 3
The Angle Addition Postulate says that
m ∠ABD + m ∠DBC = m ∠ABC hence
m ∠ABD + m ∠DBC = 90° and therefore, the two
angles must be complementary.
Parallel and Perpendicular Lines (Revised 2010)
Page 11 of 22
Examples:
In the figure, AB ⊥ EC , FB ⊥ BD and m ∠FBA =
34°.
1.
Find m ∠EBF , m ∠ABD and m ∠DBC .
{ m ∠EBF = 56°, m ∠ABD = 56°, m ∠DBC = 34° }
2.
Which theorems did you use to find your answers?
{ If 2 sides of adj. ∠s are ⊥ ,then the ∠s are
comp.}
3.
Using the diagram above, complete the following
proof:
A
D
F
C
B
E
Given: AB ⊥ EC , FB ⊥ BD
Prove: ∠EBF ≅ ∠ABD
Statements
1) Given
1) AB ⊥ EC , FB ⊥ BD
2) ∠ABD is complementary to ∠FBA
3) ∠EBF is complementary to ∠FBA.
4) ∠EBF ≅ ∠ABD
•
Reasons
2) 2 sides of adj. ∠s are ⊥ , then ∠s are
comp.
3) 2 sides of adj. ∠s are ⊥ , then ∠s are
comp.
4) ≅ comp. Th.
Proving Lines Are Parallel - Special Angle Pairs Converses (Postulate and Theorems)
Corresponding Angles Converse: If two lines are cut by a transversal so that
corresponding angles are congruent, then the lines are parallel. {CA Post. Converse}
•
Alternate Interior Angles Converse: If two lines are cut by a transversal so that
alternate interior angles are congruent, then the lines are parallel. {AIA Th. Converse}
•
Consecutive Interior Angles Converse: If two lines are cut by a transversal so that
consecutive interior angles are supplementary, then the lines are parallel. {CIA Th.
Converse}
•
Alternate Exterior Angles Converse: If two lines are cut by a transversal so that
alternate exterior angles are congruent, then the lines are parallel. {AEA Th. Converse}
Unit 3
Parallel and Perpendicular Lines (Revised 2010)
Page 12 of 22
The following are proofs of the converse theorems: they all use the diagram below.
2
1
4
3
m
6
5
7
n
8
Given: ∠5 ≅ ∠4
Statements
1)
2)
3)
4)
∠5 ≅ ∠4
∠1 ≅ ∠4
∠1 ≅ ∠5
m  n {proof of: AIE Th. Converse}
Reasons
1)
2)
3)
4)
Given
V. A. Th.
Substitution
C. A. Post. Converse
Given: ∠5 is supplementary to ∠3
Statements
1)
∠5 is supplementary to ∠3
Reasons
1) Given
2) ∠1 is supplementary to ∠3
2) L. P. Post.
3) ∠1 ≅ ∠5
4) m  n {proof of: CIA Th. Converse}
3) ≅ Supp. Th.
4) C. A. Post. Converse
Given: ∠1 ≅ ∠8
Statements
1)
2)
3)
4)
Unit 3
∠1 ≅ ∠8
∠1 ≅ ∠4
∠8 ≅ ∠4
m  n {proof of: AEA Th. Converse}
Reasons
1)
2)
3)
4)
Given
V. A. Th.
Substitution
C. A. Post. Converse
Parallel and Perpendicular Lines (Revised 2010)
Page 13 of 22
Examples: Given the information, what other information would ensure the
lines m and n were parallel?
Discuss with students: Which theorem or postulate did you use to arrive at your
answers?
1. m ∠1= 113°
m7 =°
{ m ∠1, 4, 5, 8 =°
113 and m ∠2,3, 6,
67 }
2. m ∠7 = 47°
2
1
{ m ∠1, 4, 5, 8 =
133° and m ∠2,3, 6, 7 =°
47 }
4
What value of x will ensure lines
m
and
are parallel? Find the angle measures.
3. m ∠8
= (2x + 1)°, m ∠4
= (3x-61)°
{ x = 62 ,
125° and m ∠2,3, 6, 7 =°
55 }
m ∠1, 4, 5, 8 =
4. m ∠5
= (2x + 1)°, m ∠2
= (3x-61)°
{ x = 48 ,
m ∠1, 4, 5, 8 =°
97 and m ∠2,3, 6, 7 =°
83 }
3
n
m
5
8
7
n
6
t
More Line Theorems
o If two lines are parallel to the same line, they are parallel to each other.
n
l
m
If m  n and n  l then m  l .
o In a plane, if two lines are perpendicular to the same line, they are parallel to each other.
m
l
n
If m ⊥ l and n ⊥ l then m  n .
These provide the perfect opportunity to review converse statements and
biconditional statements.
A simple exercise might include writing the Theorems and their converses as
biconditional statements.
Example: Two lines cut by a transversal are parallel if and only if they form
congruent corresponding angles.
Unit 3
Parallel and Perpendicular Lines (Revised 2010)
Page 14 of 22
Syllabus Objective: 3.5 – The student will construct parallel and perpendicular lines.
(requires supplemental material)
Acceptable methods of construction include patty paper or
compass/straightedge constructions.
PERPENDICULAR LINES (perpendicular to a given line from a point on the line)
Compass/straightedge:
A
C
B

Construct a perpendicular to AB through C.
Using C as the center, draw two arcs with equal radii,

intersecting AB at two points (D and E).
A
D
C
E
B
Using points D and E as centers, draw arcs with equal radii
intersecting at a point (F). Note, the radii used must be greater
than the length of DC or CE .
A
D
F
C
E
B
  
Draw CF . CF ⊥ AB
F
A
D
C
E
B
*Review concept by cutting frames out and having students reorder them (with or without
diagrams).
Unit 3
Parallel and Perpendicular Lines (Revised 2010)
Page 15 of 22
Patty



Paper:
Fold or draw a line on a patty paper.
Place a point on the line.
Fold the patty paper so that the crease goes through the point and one side of the line
lies on top of the other side of the line.
PERPENDICULAR LINES (perpendicular to a given line from a point NOT on the line)
Compass/straightedge:

Construct a perpendicular to AB through C.
B
A
C

Using C as the center, draw an arc intersecting AB at two
points (D and E). Note, the radii used must be greater than the

distance from C to AB .
D
E
B
A
C
Using points D and E as centers, draw arcs with equal radii that
intersect at a point (F) on the opposite side of the line. Note,
the radii need not be changed from the previous setting.
F
D
E
B
A
C
Unit 3
Parallel and Perpendicular Lines (Revised 2010)
Page 16 of 22
  
Draw CF . CF ⊥ AB .
F
D
E
A
B
C
*Review concept by cutting frames out and having students reorder them (with or
without diagrams).
Patty



Paper:
Fold or draw a line on a patty paper.
Place a dot on your patty paper to represent the given point. (not on the line)
Fold the line on top of itself so that the fold passes through the given point.
PARALLEL LINES (parallel to a given line through any point):
The easiest method for students to understand is the application of the
postulate: “In a plane, if two lines are perpendicular to the same line, they are
parallel to each other.” Students are able to perform a combination of the
perpendicular line constructions; 1st from a point NOT on the line, 2nd from a
point on the line.
Syllabus Objective: 3.6 – The student will compare strategies for determining the slope of
a line.
Slope can be introduced as a rate of change. {ex.: miles per hour, dollars per hour, feet
per second, etc…}
Looking at the numeric sequence 4, 7, 10, …, we notice a constant increase of 3 (called the
“common difference”), which is compared to a slope of 3. Any given term x of this
sequence can be found by the expression 3x + 1.
Unit 3
Parallel and Perpendicular Lines (Revised 2010)
Page 17 of 22
Similarly, a table of values can be used to demonstrate the slope idea.
n 1 2 3 4 …
t 5 4.5 4 3.5 …
1
Again there is a constant decrease of .5 or a slope of − . The nth term of the sequence
2
1
1
is found by t =
− n +5 .
2
2
rise
, it would be
run
∆y change in y
vertical change
helpful to introduce the concepts of
,
or
to
∆x change in x
horizontal change
Most students are familiar with slope being referred to as
encourage greater understanding and linkage with higher level
science/mathematic concepts.
Determining the slope of a line:
 Given the equation of a line → solve for y. (slope-intercept form)
=
y mx + b , m is the slope.
 Given two points → use slope formula.
y −y
vertical change
=
m = 2 1
horizontal change x 2 − x1
 Given a graph of the line or a graph of the points → use lattice points.
(count
vertical change
rise
or
)
run
horizontal change
Example:
6
Allows linkage to distance
formula and Pythagorean
Theorem!
5
4
3
rise of 4 units
2
1
-2
Slope of segment:
run of 3 units
2
4
−4
4
or
3
−3
6
-1
Unit 3
Parallel and Perpendicular Lines (Revised 2010)
Page 18 of 22
Syllabus Objective: 3.7 – The student will analyze slopes in a coordinate plane.
In the figure,
6
j k
because the slope of
2
. The lines j and k
3
are both perpendicular to line m because
their slopes are “opposite reciprocals” or, in
other words, the product of their slopes is 1.
2
Slope of j and k:
3
3
Slope of m: −
2
2  3
•  −  =−1
3  2
both lines are equal to
4
2
-5
5
j
-2
k
m
-4
Parallel Lines in the Coordinate Plane
o In the coordinate plane, two non-vertical lines are parallel if and only if they have equal
slopes. All vertical lines are parallel.
o Slope is the rate of change in y to the change in x.
Perpendicular Lines in the Coordinate Plane
o In the coordinate plane, two lines are perpendicular if and only if the product of their
slopes is -1. Vertical and horizontal lines are perpendicular.
o Slopes of perpendicular lines are “opposite reciprocals.”
Examples: Choose if the line pairs are parallel, perpendicular or neither.
1)
x = −1
1
3) =
y
x +3
y = −1
2
2y − x =
10
Perpendicular
Parallel
2)
4x + 3y =
6
3x + 4 y =
12
Neither
Unit 3
4) 6x − 2y =
8
1
3
Perpendicular
y =
− x − 11
Parallel and Perpendicular Lines (Revised 2010)
Page 19 of 22
Syllabus Objective: 3.8 – The student will formulate strategies to write the equation of a
line given certain data.
Review concepts:
Formats for equations of lines
o
Standard Form: Ax + By =
C
o
{A must be a whole number}
Slope-intercept Form:=
y mx + b
o
{m = slope, b = y-intercept}
Point-slope Form: y − y1= m (x − x1 )
{m = slope, (x1, y1 ) = x and y coordinates of a given point}
o
Through the point (h, k):
•
Horizontal line: y = k
•
Vertical line: x = h
Examples: (all answers given in slope-intercept form)
6
j
4
k
2
-5
5
-2
m
-4
1.
Write the equations of the lines j,
k
and
m in slope-intercept form.
2.
2
2
3
Line k :=
Line m : y =
x +2
y
x −1
− x +4
3
2
3
Write the equation of the line perpendicular to j through (-3, 0).
3.
3
9
2
2
Write the equation of the line parallel to
Line
j :=
y
y =
− x −
m through (0, 2).
3
2
y =
− x +2
Unit 3
Parallel and Perpendicular Lines (Revised 2010)
Page 20 of 22
6
4
B (-3, 2 )
2
A (-5, 0)
C (1 , 2)
D (1 , 0)
-
5
5
E (0, -1 )
2
4
6
4.
5.
6.
7.
8.
Unit 3
 

Write the equations of lines AC , CE , and BD in slope-intercept form.



1
1
1
5
AC =
:y
x+
BD : y =
− x +
CE :=
y 3x − 1
2
2
3
3

Write the equation of the line parallel to BD through A.
1
5
y =
− x −
2
2

Write the equation of the line perpendicular to BD through A.
=
y 2x + 10

Write the equation of the line parallel to AC through B.
1
=
y
x +3
3

Write the equation of the line perpendicular to CE through E.
1
y =
− x −1
3
Parallel and Perpendicular Lines (Revised 2010)
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Sample Nevada High School Proficiency Exam question(s): Taken from the 2009
Instructional Materials for the NHSPE provided by the Nevada Department of Education
In which graph do line m and p represent a system of equations that has no solution?
Unit 3
Parallel and Perpendicular Lines (Revised 2010)
Page 22 of 22