Download Chapter 3 – Parallel and Perpendicular Lines

CHAPTER 3 – PARALLEL AND PERPENDICULAR LINES
In this chapter we address three Big IDEAS:
1) Using properties of parallel and perpendicular lines
2) Proving relationships using angle measures
3) Making connections to lines in algebra
Section:
3 – 1 Identify Pairs of Lines and Angles
Essential
Question
What angles pairs are formed by two lines and a transversal?
Warm Up:
Key Vocab:
Parallel Lines
Two lines that do not intersect and are
coplanar
l‖m
Skew Lines
Lines that do not intersect and are NOT
coplanar
Parallel Planes
Two planes that do not intersect
Student Notes  Geometry  Chapter 3 – Parallel and Perpendicular Lines  KEY
Page #1
Plane R || Plane Q
Transversal
A line that intersects two or more
coplanar lines at different points
line t is a transversal
Corresponding
Angles
Two angles that are formed by two lines
and a transversal and occupy
corresponding positions
1 and 2 are
corresponding angles
Alternate
Interior Angles
Two angles that are formed by two lines
and a transversal and lie between the
two lines and on opposite sides of the
transversal.
1 and 2 are
interior angles
Alternate
Exterior Angles
Two angles that are formed by two lines
and a transversal and lie outside the two
lines and on opposite sides of the
transversal.
1 and 2 are
Alternate exterior angles
Consecutive
Interior Angles
Two angles that are formed by two lines
and a transversal and lie between the
two lines and on the same side of the
transversal.
Also called same-side interior angles.
Student Notes  Geometry  Chapter 3 – Parallel and Perpendicular Lines  KEY
Page #2
1 and 2 are
consecutive interior angles
Postulates:
If
Parallel Postulate
Then
there is a line and a point not on the line,
If
there is exactly one line through the point
parallel to the given line.
Perpendicular Postulate
Then
there is a line and a point not on the line,
there is exactly one line through the point
perpendicular to the given line.
Student Notes  Geometry  Chapter 3 – Parallel and Perpendicular Lines  KEY
Page #3
Show:
Ex 1: Think of each segment in the figure as part of a line. Which line(s) or plane(s) appear
to fit the description?
a. Line(s) parallel to ED and containing point C. GC
b. Line(s) skew to ED . BC , AG , AF , BH
c. Line(s) perpendicular to ED . CD , GE , FE , HD
d. Plane(s) parallel to plane ABH. plane CDE
Ex 2: The figure shows a sing set on a playground.
a. Name a pair of perpendicular lines.
Sample Answer: DH  HI
b. Name a pair of parallel lines.
Sample Answer: AB LM
c. Is DH perpendicular to LM ?
Explain. No, the lines are skew.
Ex 3: Identify all pairs of angles of the given type.
a. Corresponding
<1 and <5; <2 and <6;
<3 and <7; <4 and <8.
b. Alternate interior <3 and <6; <4 and <5
c. Alternate exterior <1 and <8; <2 and <7
d. Consecutive interior
<3 and <5; <4 and <6
Closure:

What angle pairs are formed by two lines and a transversal?
Angle pairs formed by three intersecting lines include corresponding, alternate interior,
consecutive interior, and alternate exterior angles.
Student Notes  Geometry  Chapter 3 – Parallel and Perpendicular Lines  KEY
Page #4
Section:
3– 2 Use Parallel Lines and Transversals
Essential
Question
How are corresponding angle pairs and alternate interior angle pairs
related when two parallel lines are cut by a transversal?
Warm Up:
Postulates and Theorems:
Corresponding Angles Postulate
Then
If
two parallel lines are cut by a transversal,
l || m
the pairs of corresponding angles are
congruent.
2  6
l
m
Student Notes  Geometry  Chapter 3 – Parallel and Perpendicular Lines  KEY
Page #5
Alternate Interior Angles Theorem
Then
If
two parallel lines are cut by a transversal,
the pairs of alternate interior angles are
congruent.
4  5
l || m
l
m
Alternate Exterior Angles Theorem
then
If
two parallel lines are cut by a transversal,
the pairs of alternate exterior angles are
congruent.
1  8
l || m
l
m
Consecutive Interior Angles Theorem
then
If
two parallel lines are cut by a transversal,
l || m
the pairs of consecutive interior angles are
supplementary.
m3  m5  180
l
m
Student Notes  Geometry  Chapter 3 – Parallel and Perpendicular Lines  KEY
Page #6
Show:
Ex 1: The measure of three of the numbered angles is 55o. Identify the angles. Explain
your reasoning.
If the lines are parallel: 6, 3, 7;
m6  55 by the Corresponding Angles
Postulate; m3  55 and m7  55 by the
Vertical Angles Congruence Theorem.
Ex 2:
Find the value of x. 165 by the Corresponding Angles Postulate
Ex 3: Prove that if two parallel lines are cut by a transversal, then the exterior angles on
the same side of the transversal are supplementary.
Given: p q
Prove: 1 and 2 are supplementary
Steps
1.
p q
Reasons
1. Given
2. m1  m3  180
2. Linear Pair Post.
3. 2  3
3. Corresponding Ang’s Post.
4. m2  m3
4. Def of Cong Ang’s
5. m1  m2  180
5. Substitution
6. 1 and 2 are supplementary
6. Def. of Supplementary Ang’s
Try:
Student Notes  Geometry  Chapter 3 – Parallel and Perpendicular Lines  KEY
Page #7
Ex 4: Prove the Alternate Interior Angles Theorem: if two parallel lines are cut by a
transversal, then the alternate interior angles are congruent.
Given: p q
2
3
Prove: 2  3
Steps
p q
1.
Reasons
1. Given
2. 1  3
2. Vert. Ang’s Congr. Thm.
3. 1  2
3. Corresponding Ang’s Post.
4. 2  3
4. Transitive Prop.
5.
5.
6.
6.
Closure:

When are pairs of corresponding, alternate interior, and alternate exterior angles
congruent?
Pairs of corresponding, alternate interior, and alternate exterior angles are congruent only
when two parallel lines are cut by a transversal

When are consecutive interior angles supplementary?
Consecutive interior angles are supplementary only when two parallel lines are cut by a
transversal.
Student Notes  Geometry  Chapter 3 – Parallel and Perpendicular Lines  KEY
Page #8
Section:
3 – 3 Prove Lines are Parallel
Essential
Question
How do you prove that lines are parallel?
Warm Up:
Key Vocab:
Paragraph
Proof
A type of proof written in paragraph form
Postulates and Theorems:
IF
Corresponding Angles Postulate Converse
then
two lines are cut by a transversal so the
corresponding angles are congruent,
the lines are parallel.
2  6
l || m
l
m
Student Notes  Geometry  Chapter 3 – Parallel and Perpendicular Lines  KEY
Page #9
Alternate Interior Angles Theorem Converse
then
If
two lines are cut by a transversal so the
alternate interior angles are congruent
the lines are parallel.
4  5
l || m
l
m
Alternate Exterior Angles Theorem Converse
then
If
two lines are cut by a transversal so the
alternate exterior angles are congruent
the lines are parallel.
1  8
l || m
l
m
Consecutive Interior Angles Theorem Converse
then
If
two lines are cut by a transversal so the
consecutive interior angles are supplementary
m3  m5  180
the lines are parallel.
l || m
l
m
Student Notes  Geometry  Chapter 3 – Parallel and Perpendicular Lines  KEY
Page #10
If
Transitive Property of Parallel Lines
then
two lines are parallel to
the same line,
they are parallel to each
other.
a
b
c
If a||b and b||c, then a||c.
Show:
Ex 1: Find the value of y that makes a || b. 23 by the alternate exterior angles converse
Ex 2: Can you prove that lines a and b are parallel? Justify your answer.
a. Yes, Alt. Ext.  ' s Conv
b. Yes, Corr.  ' s Conv.
c. No
Student Notes  Geometry  Chapter 3 – Parallel and Perpendicular Lines  KEY
Page #11
Ex 3: Prove that if 1 and 4 are supplementary then a || b.
Given: 1 and 4 are supplementary
Prove: a || b
Steps
Reasons
1. 1 and 4 are supplementary
1. Given
2. m1  m4  180
2. Def of Supplementary Ang’s
3.
m1  m2  180
m3  m4  180
3. Linear Pair Post.
4. m1  m2  m3  m4  360
4. Add. Prop. Of Eq
5. m1  m4  m2  m3  360
5. Commutative Prop. Of Eq
6. 180  m2  m3  360
6. Substitution
7. m2  m3  180
7. Subtr Prop. Of Eq
8. a || b
8. Consec Int Ang’s Converse
Ex 4: Prove the Alternate Interior Angles Theorem Converse in two-column form.
Given: 2  3
Prove: a || b
2
Steps
Reasons
1. 2  3
1. Given
2. 1  2
2. Vert. s  Theorem
3. 1  3
3. Transitive Property
4. a b
4. Corr.  ' s Postulate Converse
Ex 5: Prove the Alternate Interior Angles Theorem Converse in paragraph form.
Student Notes  Geometry  Chapter 3 – Parallel and Perpendicular Lines  KEY
Page #12
Given: 2  3
Prove: a || b
2
2
Angles 1 and 2 are vertical angles, and by the vertical angles congruence theorem,
congruent. Since we know that 2  3 , we can use the transitive property to show
1  3 . Finally, because these two angles are corresponding, we can use the
corresponding angles postulate converse to show a || b.
Student Notes  Geometry  Chapter 3 – Parallel and Perpendicular Lines  KEY
Page #13
Ex 6: Write a paragraph proof.
Given: a || b; 1  3 .
Prove c || d.
It is given that a || b, so 1  2 by Alternate Exterior Angle Theorem. Since 1  3 , it
follows that 2  3 by Transitive Property of Congruence. Therefore, c || d by the
Alternate Interior Angles Converse.
Student Notes  Geometry  Chapter 3 – Parallel and Perpendicular Lines  KEY
Page #14
Closure:

What does the Corresponding Angles Postulate prove? What do you need to know
in order to use this postulate?
The Corresponding Angles Postulate proves that when you have parallel lines, pairs of
corresponding angles are congruent. In order to use the Corresponding Angles
Postulate, you need to know that two lines are parallel.

What does the Corresponding Angles Postulate Converse say? What do you need to
know in order to use Corresponding Angles Postulate Converse?
The Corresponding Angles Postulate Converse says that when you have congruent
corresponding angles, the lines creating them must be parallel. In order to use this
postulate, you must know that a pair of corresponding angles is congruent.

Name FIVE ways to prove that lines are parallel.
1. The Corresponding Angles Postulate Converse
2. The Alternate Interior Angles Theorem Converse
3. The Alternate Exterior Angles Theorem Converse
4. The Consecutive Interior Angle Theorem Converse
5. The Transitive Property of Parallel Lines
Student Notes  Geometry  Chapter 3 – Parallel and Perpendicular Lines  KEY
Page #15
Section:
3 – 4 Find and use Slope of Lines
Essential
Question
How do you find the slope of a line given the coordinates of two
points on the line?
Warm Up:
Key Vocab:
Slope
The slope m of a nonvertical line is the ratio of the
vertical change (the rise) to the horizontal change (the
run) between any two points on the line
m
y2  y1 rise

 rate of change
x2  x1 run
Special Cases
(3,6)
(3,-2)
Vertical Lines
Horizontal Lines
Undefined or No Slope
Zero Slope
m
62 8
 
33 0
(-2,4)
(5,4)
m
Student Notes  Geometry  Chapter 3 – Parallel and Perpendicular Lines  KEY
44
0

0
2  5 7
Page #16
Slopes of Parallel Lines
In a coordinate plane, two non-vertical lines are parallel if
and only if they have the same slope.
m1 = m2
Any two vertical lines are parallel.
Slopes of Perpendicular Lines
In a coordinate plane, two non-vertical lines are
perpendicular if and only if the product of their slopes is -1.
m1(m2) = -1
Parallel and perpendicular slopes will be negative
reciprocals.
Horizontal lines are perpendicular to vertical lines.
Show:
Ex 1: Find the slope of each line. Which
lines are parallel?
Ex 2: Find the indicated slope.
a.) the line containing points (0, 3) and (5, 2)
m
y2  y1 2  3 1


x2  x1 5  0 5
b.) the line perpendicular to the line
containing (0, 3) and (5, 2)
m 
5
1
c.) the line parallel to the line containing
(0, 3) and (5, 2)
ma  
1
4
mb  mc  
1
3
mb || mc
m|| 
1
5
Ex 3: The population of a small town was 32,150 in 1990. A census in 2002 showed that
the population was 22,550. Find the average rate of change in population per year from
1990 to 2002. Assuming this rate remains constant, what should the population be in 2003?
y2  y1 22500  32150 9600


 800
x2  x2
2002  1990
12
22,500  800  21, 700 people
Student Notes  Geometry  Chapter 3 – Parallel and Perpendicular Lines  KEY
Page #17
Ex 4: A candle is lit at the center of the table at the beginning of a dinner party. Its height
decreases by 1.75 inches every 25 minutes. At the beginning of the dinner, the candle was 8
inches tall.
a. Make a table showing the height of the candle at 25 minute intervals.
Time (min)
Height (in)
0
8
25
6.25
50
4.5
75
2.75
b. Write a fraction that represents the decrease in height for each minute the candle
burns. What does the numerator represent?

0.07 height

; the height of the candle.
1
min
c. The stick candle is replaced with a thinner candle when dessert is served. It burns
with a slope of -0.5, starting at a height of 5 inches. Use a graph to compare the way
the candles burn. Which graph is steeper?
Answer:
The second graph is steeper which
means it burns faster
Closure:

What is the slope of a vertical line?
Vertical lines have undefined or no slope.

What is the slope of a horizontal line?
Horizontal lines have zero slope.
Student Notes  Geometry  Chapter 3 – Parallel and Perpendicular Lines  KEY
Page #18
Section:
3– 5 Write and Graph Equations of Lines
Essential
Question
How do you write an equation of a line?
Warm Up:
Key Vocab:
A linear equation written in the form y = mx + b where m is the
Slope-intercept Form slope and b is the y-intercept.
Best form for graphing.
Standard form
Point-slope Form
Vertical Line
Horizontal Line
A linear equation written in the form Ax + By = C, where A, B and
C are integers and A and B are not both zero.
A linear equation written in the form ( y  y1 )  m( x  x1 ) , where m
is the slope and ( x1 , y1 ) is a coordinate.
x  a where a is the x-intercept
Vertical lines have undefined or no slope
y  b where b is the y-intercept
Horizontal lines have zero slope
Student Notes  Geometry  Chapter 3 – Parallel and Perpendicular Lines  KEY
Page #19
Show:
Ex 1: Write an equation of the line in
slope-intercept form. y = 4x - 3
Ex 2: Graph 2x – 3y = -12.
y
2
x4
3
Ex 3: The graph shows the cost of having cable television installed in your home. Write
an equation of the line. Explain the meaning of the slope and the y-intercept of the line.
y = 40x + 80
The slope is the cost per month m 
cost
,
month
and the y-intercept is the initial cost.
Answer: y 
2
x4
3
Student Notes  Geometry  Chapter 3 – Parallel and Perpendicular Lines  KEY
Page #20
Ex 4: Write an equation of the line passing through the point (2, -3) that is parallel to the
line with the equation y = 6x + 4.
m6
m||  6
y  y1  m( x  x1 )
y  3  6( x  2)
y  3  6 x  12
y  6 x  15
Try:
Ex 5: Write an equation of the line a passing through the point (3, -4) that is perpendicular
to the line with the equation y = -½x – 1.
m
1
2
m  2
y  y1  m( x  x1 )
y  4  2( x  3)
y  4  2x  6
y  2 x  10
Closure:

What are the FIVE forms of a linear equation?
1. Standard Form: Ax  By  C
2. Slope-Intercept Form: y  mx  b
3. Point-Slope Form: y  y1  m( x  x1 )
4. Vertical Line: x  a
5. Horizontal Line: y  b
Student Notes  Geometry  Chapter 3 – Parallel and Perpendicular Lines  KEY
Page #21
Section:
3 – 6 Prove Theorems about Perpendicular Lines
Essential
Question
How do you find the distance between a point and a line?
Warm Up:
Key Vocab:
Distance from
a Point to a
Line
The length of the perpendicular
segment from a point to a line.
Theorems:
If
two lines intersect to
form a linear pair of
congruent angles,
Then
the lines are
perpendicular.
If
Then
two lines are
perpendicular,
they intersect to form
four right angles.
If 1  2 , then g  h.
If a  b, then 1, 2, 3, and 4 are right
angles.
If
Then
two sides of two
adjacent acute angles
are perpendicular,
the angles are
complementary.
If BA  BC , then 1 and 2 are
complementary.
Student Notes  Geometry  Chapter 3 – Parallel and Perpendicular Lines  KEY
Page #22
Perpendicular Transversal Theorem:
If
Then
a transversal is perpendicular to one of two
parallel lines,
it is perpendicular to the other.
h || k and j  h
In a plane, if
jk
Lines Perpendicular to a Transversal Theorem:
Then
two lines are perpendicular to the same line,
they are parallel to each other.
If m  p and n  p,
m n.
Show:
Ex 1: In the figure, 1 and 2 are congruent. What can you conclude about m2 ?
m2  90
Ex 2: Determine which other lines, if any, must be perpendicular. Explain your reasoning.
By the Corr. Angles Post Conv., a || b .
So, b  c Perp. Transversal Theorem
Ex 3: Is a || b ? Is b  c ? Explain your reasoning.
Yes, a || b by the Lines Perpendicular to a Transversal
Theorem
Yes, b  c : c || d by the Lines Perpendicular to a Transversal
Theorem, so b  c by the Perpendicular Transversal Theorem
Student Notes  Geometry  Chapter 3 – Parallel and Perpendicular Lines  KEY
Page #23
Ex 4: Prove the Perpendicular Transversal Theorem
1
Given: h || k ; j  h
2
Prove: j  k
Steps
Reasons
1. h || k ; j  h
1. Given
2. m1  90
2. Def of Perp. Lines
3. 1  2
3. Alt. Int. Angles Thm
4. m1  m2
4. Def. of Congr. Ang’s
5. m2  90
5. Def of Rt. Ang’s
6.
jk
6. Def of Perp. Lines
Ex 5: What is the distance between the two parallel sides of this table top?
Graphical Method:
d
m 
3
4
m 
4
3
 3  1   0  3
2
2
 16  9
 25
5
Algebraic Method:
3
4
m 
4
3
3
y1  x  3
4
3
y2  1  ( x  3)
4
3
9
y2  1  x 
4
4
3
13
y2  x 
4
4
1. m 
4
2. (0,3), m 
3
4
y 3 
( x  0)
3
4
y 
x3
3
3.
y  y2
4
3
13
x3 x
3
4
4
4
y 
(3)  3
13 3
4
3  x  x
3
4 4
3
y  1
25 25

x
4 12
x3
4.
d
 3  1   0  3
2
2
 16  9
 25
5
Closure:
Student Notes  Geometry  Chapter 3 – Parallel and Perpendicular Lines  KEY
Page #24

Compare and contrast the Perpendicular Transversal Theorem and the Lines
Perpendicular to a Transversal Theorem.
The Perpendicular Transversal Theorem is used to prove that two lines are perpendicular,
while the Lines Perpendicular to a Transversal Theorem is used to prove that two lines are
parallel.
The Perpendicular Transversal Theorem requires one line to be perpendicular to one of two
parallel lines, whereas the Lines Perpendicular to a Transversal Theorem requires two lines
to be perpendicular to the same line.

Summarize the method for algebraically finding the distance between two lines
1. Find the slope of the parallel linesFind the slope of a perpendicular line.
2. Find the equation of a perpendicular line. (Use the perpendicular slope
and choose a point on one of the two parallel lines)
3. Use a system of equations to find the intersection of the perpendicular line
and the second parallel line. (set the first parallel equation equal to the
perpendicular equation then solve for x and y.)
4. Use the distance formula to find the distance. (plug the chosen point and the
intersection point into the distance formula)
Student Notes  Geometry  Chapter 3 – Parallel and Perpendicular Lines  KEY
Page #25