Name: _______________________________________________ Date: ____________________
Chapter 3
Parallel
and
Perpendicular Lines
Sections Covered:
3.1 Identify Pairs of Lines and Angles
3.2 Use Parallel Lines and Transversals
3.3 Prove Lines are Parallel
3.4 Find and Use Slopes of Lines
3.5 Write and Graph Equations of Lines
The student will use the relationships between angles formed by two lines cut by a transversal to
a)
determine whether two lines are parallel;
b)
verify the parallelism, using algebraic and coordinate methods as well as
deductive proofs; and
c)
solve real-world problems involving angles formed when parallel lines are cut
by a transversal.
SOL G.2
The student will use pictorial representations, including computer software, constructions, and coordinate methods,
to solve problems involving symmetry and transformation. This will include
a)
investigating and using formulas for finding distance, midpoint, and slope;
b)
applying slope to verify and determine whether lines are parallel or
perpendicular;
SOL G.3
Unit 3 Syllabus: Ch 3 Parallel and Perpendicular Lines
Block Date
Topic
Homework
16
B 10/16
A 10/17
17
B 10/20
A 10/21
B 10/22
A 10/23
B 10/24
A 10/27
18
19
20
3.1 Identifying Lines and Angles
3.2 Angle Relationships and Parallel
Lines
3.3 Proving Lines Parallel
Quiz 3.1-3.3
3.4 Slope of Lines
3.5 Writing Equations of Lines
Worksheet: 3.1 and 3.2
Identifying Lines and
Angle Relationships
Worksheet: 3.3 Proving
Lines Parallel
Khan Academy
Worksheet: 3.4 Slope of
Lines
3.5 Writing Equations
of Lines
Review Worksheet—
Separate from packet
Review Worksheet #1
B 10/28
Review Ch 1 and 2 for Quarter Test
A 10/29
21
B 10/30
Quarter 1 Benchmark Test
A 10/31
1
B 11/5
Review Day Ch 3
Review Worksheet #2
A 11/6
2
B 11/7
Ch 3 Test
Khan Academy
A 11/8
***Syllabus subject to change due to weather, pep rallies, illness, etc
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2
Notes 3.1 & 3.2: Identifying Lines and Angles and Angle Relationships
_______________________________________________________
PARALLEL
PERPENDICULAR
LINES-
LINES-
SKEW
LINES-
DEFINITION
A
K
B
J
C
D
EXAMPLE
L
F
G
E
N
H
PLANES-
A
B
C
D
P
M
F
E
G
H
PLANES-
DEFINITION
A
B
C
D
EXAMPLE
J
L
F
E
K
G
N
H
P
M
Think of each segment in the diagram as part of a line. Which line(s) or plane(s) appear to
fit the description?
1. Line(s) parallel to AB
2. Line(s) perpendicular to BF
3. Line(s) skew to CD and containing point E
4. Plane(s) perpendicular to plane ABE
5. Plane(s) parallel to plane ABC
3
This table defines the five types of angles by their location.
It also states the specific relationships they have when the transversal cuts through
parallel lines.
Type:
Location:
Picture:
Theorem: If 2 parallel lines are cut by a transversal, then
___________________________ are _______________.
Type:
Location:
Picture:
Theorem: If 2 parallel lines are cut by a transversal, then
___________________________ are _______________.
Type:
Location:
Picture:
Theorem: If 2 parallel lines are cut by a transversal, then
___________________________ are _______________.
Type:
Location:
Picture:
Theorem: If 2 parallel lines are cut by a transversal, then
___________________________ are _______________.
Type:
Location:
Picture:
Theorem: If 2 parallel lines are cut by a transversal, then
___________________________ are _______________.
4
Classify each angle pair as corresponding, alternate interior, alternate exterior, consecutive
interior, or consecutive exterior.
a) ∠1 and ∠9 ____________________
b) ∠8 and ∠13 ____________________
c) ∠6 and ∠16 ____________________
d) ∠4 and ∠10 ____________________
e) ∠8 and ∠16 ____________________
f) ∠10 and ∠13 ____________________
Discovery: Lines l and m are parallel. Note: Parallel lines are distinguished by a matching
set of arrows on the lines that are parallel. Find the measure of the missing angles by using
transparent paper. Then, let’s go back and fill in the theorems.
l
115°
m
x°
Key Question: If x = 115°, is it possible for y to equal 115°?
y°
z°
For t he following diagrams, state the type of angles that are given, state their relationship,
and then find x.
1.
2.
3.
5
Find the missing variables.
4.
75°
5.
(5x – 10)°
(10y – 25)°
60°
(3x + 15)°
6.
(–5y – 10)°
On Your Own: For t he following diagrams, state the type of angles that are given, state
their relationship, and then find x.
1.
2.
4.
5.
3.
100°
6.
120°
3x°
(x – 10)°
(2y + 24)°
(6y – 12)°
6
Notes 3.3: Proving Lines Parallel
_______________________________________________________
Follow along and fill in the missing blanks for each theorem. Then, based on the theorem, use the
given theorem to determine if the lines are parallel or not parallel. Provide reasoning.
Corresponding Angles Converse Theorem: If 2 lines are cut by a transversal so the
corresponding angles are __________________, then the lines are __________________.
Example:
Non Example:
2
3
j
6
l
k
6
k
Alternate Interior Angles Converse Theorem: If 2 lines are cut by a transversal so the
alternate interior angles are __________________, then the lines are __________________.
Example:
Non Example:
25°
j
105°
130°
l
92°
88°
k
k
Alternate Exterior Angles Converse Theorem: If 2 lines are cut by a transversal so the
alternate exterior angles are __________________, then the lines are __________________.
Example:
Non Example:
28°
j
106°
k
l
k
137°
Consecutive Interior Angles Converse Theorem: If 2 lines are cut by a transversal so the
consecutive interior angles are __________________, then the lines are __________________.
Example:
Non Example:
H
K
96°
84°
j
k
J
I
7
Consecutive Exterior Angles Converse Theorem: If 2 lines are cut by a transversal so the
consecutive exterior angles are __________________, then the lines are __________________.
Transitive Property of Parallel Lines:
Example:
u
p
t
Is it possible to prove the lines are parallel or not parallel? If so, state the postulate or
theorem you would use. If not, state cannot be determined.
1.
2.
3.
92°
k
88°
4.
5.
l
6.
l
k
7. A
B
105°
E
8.
122°
k
75°
I
D
9.
C
H
55°
55°
l
F
58°
m
G
8
10. Find the value of x that makes l // m.
11. a. Find the value of x that makes a // b.
b. Find the value of y that makes a // c.
c. Is b // c? Why or why not?
State the postulate or theorem that supports each conclusion.
1. Given: a || b
Conclusion:  2   7
b
____________________
5
2. Given: m  4 + m  7 = 180 ____________________
Conclusion: a || b
3. Given:  4   5
Conclusion: a || b
a
1 2
7
6
8
3 4
____________________
9
Find the values of x and y. Explain your reasoning by stating the proper theorem or
postulate.
4.
5.
6.
x
130
y
65 y
x
y
x
80
x=
y=
_____________________
_____________________
x=
y=
_____________________
_____________________
x=
y=
_____________________
_____________________
Find the value of x so that n || m. State the theorem or postulate that justifies your
solution.
5x
7.
m
n
n
8.
m
5x+23
7x+13
9.
n
5x-18
m
8x-5
3x+48
x=
x=
x=
_____________________
_____________________
_____________________
Can you prove that lines p and q are parallel? If so, state the theorem or postulate that you
would use.
p
10.
p
11.
q
q
_____________________
p
12.
q
_____________________
_____________________
10
Notes 3.4 and 3.5: Writing Equations of Lines
_______________________________________________________
Equations of Lines and Slope
Slope intercept form:
Slope Formula:
Graphing and Types of Slopes: Graph the following lines.
1
4
y=  x–2
y = 2x + 4
y=–3
x=5
Acronym:
m = ______ b = ______
m = _____ b = ______
type of slope:
type of slope:
Acronym:
type of slope:
type of slope:
For each equation, rewrite in slope-intercept form and state the m & b values.
3y – 8x = 2
9x = 4y – 11
1
3x  y  6
4
m=_________b=_________
m=_________b=_________
m=_________b=_________
11
Special Types of Lines:
TYPE OF LINE
PARALLEL LINES
PERPENDICULAR LINES
DEFINITION
SLOPES OF
THESE TYPE OF
LINES
State the negative reciprocal of the given slope.
1
1. m =
2. m = –6
4
3. m = 
2
3
Find the slope of the given lines.
j passes through
m passes through
(0, 3) and (3, 1)
( –2, 7) and (–6 , 1)
Make some conclusions.
4. m = 9
k passes through
(-4, -3) & (0, 3)
Make a quick sketch to see what
parallel and perpendicular lines
look like.
12
Write the equation of a line in slope intercept form:
Steps: 1. Ask yourself “What two letters do I need to write the equation of a line?”
2. Identify which letters you need to still find.
3. If you need m, plug the points into the slope formula.
4. If you need b, plug m and an ordered pair (x, y) into the slope intercept formula
and solve for b.
5. Write the equation of a line with the new m and b.
TYPE I: Write the equation of the line that passes through the given y-intercept and given
slope.
1. m = 3, b = -3
6
7
2. m = , b = 15
TYPE II: Write the equation of the line that passes through the given point and given slope.
3. Passes through (2, 3) and slope is 5.
5. Passes through (5, -2) and slope is 0.
4. Passes through (6, -5) and slope is 
1
3
Remember: You can always check the b
by graphing. Plot the point and move by
counting the slope till you cross the y-axis.
13
Type III: Write the equation of a line given two points.
6. Passes through (4, -3) and (3, -6)
7.
TYPE IV: Write the equation of a line given two points and must be parallel or
perpendicular to another line.
8. Passes through (3, 2)
9. Passes through (4, 0)
1
Parallel to y   x  1
Perpendicular to 2x + y = 1
3
14
Practice: Are these equations parallel, perpendicular, or neither?
1
1. l: y  x  2 h: 6y  2x  12
2. q: 4 x  2y  6
w: 2x  4y  6
3
3. Which lines are //? Which are ⊥? A graph may help.
x=4
y = –4
y = 4x
15