Download Alternate Interior Angles

Chapter 3
Perpendicular and Parallel Lines
Chapter Objectives
Identify parallel lines
Define angle relationships between parallel
lines
Develop a Flow Proof
Use Alternate Interior, Alternate Exterior,
Corresponding, & Consecutive Interior Angles
Calculate slopes of Parallel and Perpendicular
lines
Lesson 3.1
Lines and Angles
Lesson 3.1 Objectives
Identify relationships between lines.
Identify angle pairs formed by a
transversal.
Compare parallel and skew lines.
Lines and Angle Pairs
Alternate Exterior Angles –
because they lie outside the
two lines and on opposite sides
Transversal
1 2
4 3
5 6
8 7
Corresponding
Angles –
because they lie
in corresponding
positions of each
intersection.
of the transversal.
Alternate Interior Angles –
because they lie inside the two lines
and on opposite sides of the
transversal.
Consecutive
Interior
Angles –
because they
lie inside the
two lines and
on the same
side of the
transversal.
Example 1
Determine the relationship between the given angles
1)
3 and 9
1)
2)
13 and 5
2)
3)
Alternate Interior Angles
5 and 15
4)
5)
Corresponding Angles
4 and 10
3)
4)
Alternate Interior Angles
Alternate Exterior Angles
7 and 14
5)
Consecutive Interior Angles
Parallel versus Skew
Two lines are parallel if they are coplanar
and do not intersect.
Lines that are not coplanar and do not
intersect are called skew lines.

These are lines that look like they intersect but do
not lie on the same piece of paper.
Skew lines go in different directions while
parallel lines go in the same direction.
Example 2
Complete the following statements using the words parallel, skew, perpendicular.
1)
Line WZ and line XY are _________.
1)
2)
Line WZ and line QW are ________.
2)
3)
perpendicular
Line TS and line ZY are __________.
6)
7)
parallel
Plane RQT and plane WQR are _________.
5)
6)
skew
Plane WQR and plane SYT are _________.
4)
5)
perpendicular
Line SY and line WX are _________.
3)
4)
parallel
skew
Line WX and plane SYZ are __________.
7)
parallel.
Parallel and Perpendicular Postulates:
Postulate 13-Parallel Postulate
If there is a line and a point not on the
line, then there is exactly one line
through the point that is parallel to the
given line.
Parallel and Perpendicular Postulates:
Postulate 14-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.
Homework 3.1
In Class

2-9
 p132-135
Homework

10-31
Due Tomorrow
Lesson 3.2
Proof and Perpendicular Lines
Lesson 3.2 Objectives
Develop a Flow Proof
Prove results about perpendicular lines
Use Algebra to find angle measure
Flow Proof
5
7
6
A flow proof uses arrows to show the flow of a
logical argument.

Each reason is written below the statement it justifies.
GIVEN:  5 and  6 are a linear pair
 6 and  7 are a linear pair
PROVE:  5   7
 5 and  6
are a linear
pair
 6 and  7
are a linear
pair
Given
1.  5 and  6 are a linear pair
 6 and  7 are a linear pair
1. Given
2.  5 and  6 are supplementary
 6 and  7 are supplementary
2. Linear Pair
Postulate
3.  5   7
3. Congruent
Linear Pair
Supplements
Postulate
Theorem
 5 and  6 are
supplementary
Given
 6 and  7 are
supplementary
Linear Pair
Postulate
 5 7
Congruent Supplements Theorem
Theorem 3.1:
Congruent Angles of a Linear Pair
If two lines intersect to form a linear
pair of congruent angles, then the lines
are perpendicular.
g
h
So g  h
Theorem 3.2:
Adjacent Angles Complementary
If two sides of two adjacent acute
angles are perpendicular, then the
angles are complementary.
Theorem 3.3: Four Right Angles
If two lines are perpendicular, then they
intersect to form four right angles.
Homework 3.2
None!
Move on to Lesson 3.3
Lesson 3.3
Parallel Lines and Transversals
Lesson 3.3 Objectives
Prove lines are parallel using
tranversals.
Identify properties of parallel lines.
Postulate 15:
Corresponding Angles Postulate
If two parallel lines are cut by a transversal, then
corresponding angles are congruent.

You must know the lines are parallel in order to assume the
angles are congruent.
1 2
4 3
5 6
8 7
Theorem 3.4:
Alternate Interior Angles
If two parallel lines are cut by a transversal, then alternate
interior angles are congruent.



Again, you must know that the lines are parallel.
If you know the two lines are parallel, then identify where the
alternate interior angles are.
Once you identify them, they should look congruent and they are.
1 2
4 3
5 6
8 7
Theorem 3.5:
Consecutive Interior Angles
If two parallel lines are cut by a transversal, then
consecutive interior angles are supplementary.


Again be sure that the lines are parallel.
They don’t look to be congruent, so they MUST be
supplementary.
1 2
4 3
180o = +
+
= 180o
5 6
8 7
Theorem 3.6:
Alternate Exterior Angles
If two parallel lines are cut by a transversal, then
alternate exterior angles are congruent.

Again be sure that the lines are parallel.
1 2
4 3
5
6
8 7
Theorem 3.7:
Perpendicular Transversal
If a transversal is perpendicular to one of two
parallel lines, then it is perpendicular to the
other.


Again you must know the lines are parallel.
That also means that you now have 8 right
angles!
Example 3
Find the missing angles for the following:
120o
60o
120o
120o
140o
140o
105o
110o
70o
105o
110o
Homework 3.3
In Class

3-6
 p146-149
Homework

8-26, 34-44 even
Due Tomorrow
Quiz Wednesday

Lessons 3.1-3.3
 Emphasis on 3.1 & 3.3
Lesson 3.4
Proving Lines are Parallel
Lesson 3.4 Objectives
Prove that lines are parallel
Recall the use of converse statements
Postulate 16:
Corresponding Angles Converse
If two lines are cut by a transversal so that
corresponding angles are congruent, then the lines
are parallel.

You must know the corresponding angles are congruent.
 It does not have to be all of them, just one pair to make the
lines parallel.
1 2
4 3
5 6
8 7
Theorem 3.8:
Alternate Interior Angles Converse
If two lines are cut by a transversal so that alternate
interior angles are congruent, then the lines are
parallel.

Again, you must know that alternate interior angles are
congruent.
1 2
4 3
5 6
8 7
Theorem 3.9:
Consecutive Interior Angles Converse
If two lines are cut by a transversal so that
consecutive interior angles are
supplementary, then the lines are parallel.

Be sure that the consecutive interior angles are
supplementary.
1 2
4 3
180o = +
+
= 180o
5 6
8 7
Theorem 3.10:
Alternate Exterior Angles Converse
If two lines are cut by a transversal so that alternate
exterior angles are congruent, then lines are
parallel.

Again be sure that the alternate exterior angles are
congruent.
1 2
4 3
5
6
8 7
Example 4
Is it possible to prove the lines are parallel?
If so, explain how.
Yes they are parallel!
Yes they are parallel!
Because
Alternate
Interior
Angles are
congruent.
Because Corresponding
Angles are congruent.
Corresponding Angles
Converse
Alternate Interior
Angles Converse
Yes they are parallel!
Because Alternate
Exterior Angles are
congruent.
Alternate Exterior
Angles Converse
No they are
not parallel!
No relationship between
those two angles.
Example 5
Find the value of x that makes the
m  n.
x = 2x – 95
(AIA)
100 = 4x – 28
(CA)
(3x + 15) + 75 = 180(CIA)
-x = –95
(SPOE)
128 = 4x
(APOE)
3x + 90 = 180
(CLT)
x = 95
(DPOE)
x = 32
(DPOE)
3x = 90
(SPOE)
x = 30
(DPOE)
Directions do not ask for reasons, I am showing you them because I am a teacher!!
Homework 3.4
In Class

1, 3-9
 p153-156
Homework

10-35, 37, 38
Due Tomorrow
Lesson 3.5
Using Properties of Parallel Lines
Lesson 3.5 Objectives
Prove more than two lines are parallel
to each other.
Identify all possible parallel lines in a
figure.
Theorem 3.11:
3 Parallel Lines Theorem
If two lines are parallel to the same line, then
they are parallel to each other.

This looks like the transitive property for parallel
lines.
If p // q and q // r, then p // r.
p
q
r
Theorem 3.12:
Parallel Perpendicular Lines Theorem
In a plane, if two lines are perpendicular to
the same line, then they are parallel to each
other.
If m  p and n  p, then m // n.
m
n
p
Finding Parallel Lines
Find any lines that are parallel and explain why.
b
a
x
y
z
125o
55o
c
Results
x // y

Corresponding Angles Converse
 Postulate 16
y // z

Consecutive Interior Angels Converse
 Theorem 3.9
x // z

3 Parallel Lines Theorem
 Theorem 3.11
b // c

Alternate Exterior Angles Converse
 Theorem 3.10
Homework 3.5
In Class

16, 19
 p160-163
Homework

8-24, 33-36, 43-51
Due Tomorrow
Lesson 3.6
Parallel Lines in the Coordinate Plane
Lesson 3.6 Objectives
Review the slope of a line
Identify parallel lines based on their
slopes
Write equations of parallel lines in a
coordinate plane
Slope
Recall that slope of a nonvertical line is a ratio of the
vertical change divided by the horizontal change.


It is a measure of how steep a line is.
The larger the slope, the steeper the line is.
Slope can be negative or positive whether or not the
lines slants up or down.
Remember that slope is often referred to as
Which really means
y2 – y1
x2 – x1
rise
run
=m
Which we find it in an equation by looking for m.
y = mx + b
Example of Slope
You are given two points:
A(1 , 2 )
B(3 , 8 )
1
Now label each point as 1 and 2.
2
Then substitute as the formula for slope tells you.
y2 – y1
x2 – x1
=
8
3
–
–
2
1
=
6
2
=
3
Postulate 17:
Slopes of Parallel Lines Postulate
In a coordinate plane, two nonvertical
lines are parallel if and only if they have
the same slope.

Any two vertical lines are parallel.
m = -1
m = -1
m = undefined
Writing an Equation in
Slope Intercept Form
You will be given

Slope
 Or at least two points so you can calculate
slope

y-intercept
slope
y = mx + b
Your final answer
should always
appear in this
form.
y-intercept
This is the point at
which the line
touches the y-axis.
Writing an Equation Given
Slope and 1 Point
For this, you will be given the slope


Or have to determine it from and equation
Or determine it from a set of two points
You will also be given 1 point through which the line passes
To solve, use your slope-intercept form to find b

y = mx+b
Plug in



Slope for m.
The x-value from your point for x.
The y-value from your point for y.
Solve for b using algebra
When finished, be sure to rewrite in slope intercept form using
your new m and b.

Leave x and y as x and y in your final equation.
Example
Example 5

p167
Write an equation of the line through the point (2,3) that has a
slope of 5.
y = mx + b
y = 5x + b
3 = 5(2) + b
3 = 10 + b
b = -7
So, your final answer is:
y = 5x - 7
Homework 3.6
WS
Due Tomorrow
Lesson 3.7
Perpendicular Lines in the
Coordinate Plane
Lesson 3.7 Objectives
Use slope to identify perpendicular lines
Write equations of perpendicular lines
Postulate 18:
Slopes of Perpendicular Lines Postulate
In a coordinate plane, two nonvertical
lines are perpendicular if and only if the
product of their slopes is –1.

Vertical and horizontal lines are
perpendicular.
Identifying Perpendicular Lines
There are two ways to identify
perpendicular lines

The product of the slopes equal –1
 (3/2)(-2/3) = -1

Or to get from one slope to the other, you
find the negative reciprocal
 Remember that reciprocal flips the number.
 Well now you flip it and make it negative!
 3/2  -2/3
Determining Perpendicular Lines
Remember there are two ways to identify
perpendicular lines

Multiply the slopes together
 If the answer is –1, they are perpendicular

Verify that the slopes are negative reciprocals of each other
 Take one of the slopes, flip it over, and make it negative. If the
answer matches the other slope, they are perpendicular.
Example (#25 p176)


y = 3x
y = - 1 /3 x – 2
 (3)(-1/3) = -1
 Perpendicular


Or 3  1/3  -1/3
Check!
Tougher Example
Need to change into
slope-intercept form.
y = mx + b
Example 4

P173
Decide whether the lines are perpendicular
Line s: 5x + 4y = 3
Line r: 4x + 5y = 2
5y = -4x + 2
y = - 4 /5 x + 2 /5
Subtract x-term
from both sides
4y = -5x + 3
Get y to be alone
by dividing off
the coefficient
y = - 5 /4 x + 3 /4
Now multiply their slopes
(-4/5)(-5/4) =
20/
20
=1
NOT Perpendicular
Homework 3.7
WS
Due Tomorrow
Test Thursday

January 29th