Transcript Parallel and Perpendicular Lines

Chapter 3

3.1
Lines and Angles
First thing we’re going to do is
travel to another dimension

First thing we’re going to do is
travel to another dimension

THE THIRD DIMENSION
First thing we’re going to do is
travel to another dimension

Once we get there we’ll discuss
Parallel, Perpendicular, and
Skew lines
Diagramed

Diagramed

Parallel lines
B
A
C
F
J
D
E
G
Coplanar lines that don’t
intersect.

Parallel lines
B
A
C
F
J
D
E
G
Diagramed

Parallel lines
B
AB, CF, EG, DJ
A
C
F
J
D
E
G
Diagramed

Parallel lines
AB, CF, EG, DJ
AD, BJ, FG, CE
B
A
C
F
J
D
E
G
Diagramed

Perpendicular lines
B
A
C
F
J
D
E
G
Intersect to make a right angle

Perpendicular lines
B
A
C
F
J
D
E
G
Intersect to make a right angle

Perpendicular lines
AB and BJ
B
A
C
F
J
D
E
G
Intersect to make a right angle

Perpendicular lines
AB and BJ
A
AB and BC
BJ and BC
B
C
F
J
D
E
G
If 2 lines are perpendicular to the same
line, are they perpendicular to each
other?

Perpendicular lines
AB and BJ
A
AB and BC
BJ and BC
B
C
F
J
D
E
G
Something perhaps that’s “gnu”

Skew lines
B
A
C
F
J
D
E
G
Something perhaps that’s “gnu”

Skew lines
B
A
C
F
J
D
E
G
Defines as lines in different planes that
are not parallel.

Skew lines
B
A
C
F
J
D
E
G
The only reason they don’t intersect is
because they are not coplanar.

Skew lines
B
A
C
F
J
D
E
G
Examples:

Skew lines
B
A
C
F
J
D
E
G
Examples:

Skew lines
AB and EJ
B
A
C
F
J
D
E
G
Examples:

Skew lines
AB and EJ
JD and FG
B
A
C
F
J
D
E
G
Examples:

Skew lines
AB and EJ
JD and FG
DG and CE
B
A
C
F
J
D
E
G
Make sure that…

Make sure that…

Given a diagram:
Make sure that…

Given a diagram:
Identify the relationship
between a pair of lines
Make sure that…

Given a diagram:
Identify the relationship
between a pair of lines.
Label lines so that the desired
relationship is shown
Complete the Got It? on page 141

Given a diagram:
Identify the relationship
between a pair of lines.
Label lines so that the desired
relationship is shown
Complete the Got It? on page 141

Use above it as a guide if you desire.
Returning to the flat
world…

Returning to the flat
world…

On a plane, when lines intersect two or
more lines at distinct points, the angles
formed at these points create special
angle pairs.
Returning to the flat
world…

Their description and location is based
upon a transversal.
Returning to the flat
world…

Their description and location is based
upon a transversal.
A line that intersects two or more lines
at distinct points.
These will break down into
interior and exterior locations.

These will break down into
interior and exterior locations.

Page 141 in your book.
These will break down into
interior and exterior locations.

Page 141 in your book.
Interior angles are found
between the 2 lines that are
intersected
These will break down into
interior and exterior locations.

Page 141 in your book.
Interior angles are found
between the 2 lines that are
intersected
As you can guess, exterior
angles are then found
outside these same lines.
Now we throw in alternate which
involves opposite sides of the transversal.

Now we throw in alternate which
involves opposite sides of the transversal.

Page 142
Names and Descriptions

The “glowing”
line is the
transversal.
Names and Descriptions

1 2
3 4
7
5 6
8
Alternate interior
angles are
nonadjacent
interior angles
found on
opposite sides of
the transversal.
3 and 6

1 2
3 4
7
5 6
8
Alternate interior
angles are
nonadjacent
interior angles
found on
opposite sides of
the transversal.
4 and 5

1 2
3 4
7
5 6
8
Alternate interior
angles are
nonadjacent
interior angles
found on
opposite sides of
the transversal.
Names and Descriptions

1 2
3 4
7
5 6
8
Alternate exterior
angles are
nonadjacent
exterior angles
found on
opposite sides of
the transversal.
1 and 8

1 2
3 4
7
5 6
8
Alternate exterior
angles are
nonadjacent
exterior angles
found on
opposite sides of
the transversal.
2 and 7

1 2
3 4
7
5 6
8
Alternate exterior
angles are
nonadjacent
exterior angles
found on
opposite sides of
the transversal.
Names and Descriptions

1 2
3 4
7
5 6
8
Same-side
interior angles
are nonadjacent
angles that line
on the same side
of the transversal.
3 and 5

1 2
3 4
7
5 6
8
Same-side
interior angles
are nonadjacent
angles that line
on the same side
of the transversal.
4 and 6

1 2
3 4
7
5 6
8
Same-side
interior angles
are nonadjacent
angles that line
on the same side
of the transversal.
Names and Descriptions

1 2
3 4
7
5 6
8
Corresponding
angles are angles
found on the same
side of the
transversal in the
same corresponding
or relative position.
1 and 5

1 2
3 4
7
5 6
8
Corresponding
angles are angles
found on the same
side of the
transversal in the
same corresponding
or relative position.
3 and 7

1 2
3 4
7
5 6
8
Corresponding
angles are angles
found on the same
side of the
transversal in the
same corresponding
or relative position.
2 and 6

1 2
3 4
7
5 6
8
Corresponding
angles are angles
found on the same
side of the
transversal in the
same corresponding
or relative position.
4 and 8

1 2
3 4
7
5 6
8
Corresponding
angles are angles
found on the same
side of the
transversal in the
same corresponding
or relative position.
Homework

Page 144 – 145
11 – 24, 30 – 35,
37 – 42
Answer the
questions,
identify the
desired
relationships.
3.2 – 3.3
Here’s what you’re
going to do…

Here’s what you’re
going to do…

1) On a sheet of notebook, darken in 2
horizontal lines a few
inches apart.
Here’s what you’re
going to do…

1) On a sheet of notebook, darken in 2
horizontal lines a few
inches apart.
2) Create a transversal that
is not perpendicular to
your 2 lines.
Here’s what you’re
going to do…

2) Create a transversal that
is not perpendicular to
your 2 lines.
3) Measure all 8 angles that
are formed by the transversal and the lines you
darkened.
Now for the thought process:

Now for the thought process:

What is special about the lines you
darkened?
Now for the thought process:

What is special about the lines you
darkened?
They are parallel
Now for the thought process:

What is special about the lines you
darkened?
They are parallel
What is special about pairs of
angles you measured?
Now for the thought process:

What is special about the lines you
darkened?
They are parallel
What is special about pairs of
angles you measured?
They are congruent
This is not a coincidence

This is not a coincidence

If a transversal intersects 2 parallel
lines:
This is not a coincidence

If a transversal intersects 2 parallel
lines:
(1)
Alternate interior angles are
congruent.
This is not a coincidence

If a transversal intersects 2 parallel
lines:
(1)
Alternate interior angles are
congruent.
(2)
Alternate exterior angles
are congruent.
This is not a coincidence

If a transversal intersects 2 parallel
lines:
(2)
Alternate exterior angles
are congruent.
(3)
Corresponding angles are
congruent.
This is not a coincidence

If a transversal intersects 2 parallel
lines:
(3)
Corresponding angles are
congruent.
(4)
Same side interior angles
are supplementary.
A postulate…

A postulate…

3.1
If a transversal intersects two
parallel lines, then same side
interior angles are supplementary
A list of theorems

3.1
If a transversal intersects two
parallel lines, then same side
interior angles are supplementary
A list of theorems

3.1
If a transversal intersects two
parallel lines, then alternate
interior angles are congruent
A list of theorems

3.2
If a transversal intersects two
parallel lines, then corresponding
angles are congruent
A list of theorems

3.3
If a transversal intersects two
parallel lines, then alternate
exterior angles are congruent.
The long way to find angle
measures.

Let m3 = 82
1
3
2
4
5 6
7 8
The long way to find angle
measures.

1
3
2
4
5 6
7 8
Let m3 = 82
- m2 = ____
- m1 = ____
- m4 = ____
Vertical angle conjecture

1
3
2
4
5 6
7 8
Let m3 = 82
- m2 = 82
- m1 = ____
- m4 = ____
Linear Pair Angle Conjecutre

1
3
2
4
5 6
7 8
Let m3 = 82
- m2 = 82
- m1 = 98
- m4 = ____
Linear Pair or Vertical Angle
Conjecture

1
3
2
4
5 6
7 8
Let m3 = 82
- m2 = 82
- m1 = 98
- m4 = 82
Now we march on to the other
point of intersection

1
3
2
4
5 6
7 8
Let m3 = 82
- m2 = 82
- m1 = 98
- m4 = 82
Now we march on to the other
point of intersection

1
3
2
4
5 6
7 8
Let m3 = 82
- m5 = ____
- m6 = ____
- m7 =____
- m8 =____
By the Same-Side Conjecture

1
3
2
4
5 6
7 8
Let m3 = 82
- m5 = ____
- m6 = ____
- m7 =____
- m8 =____
By the Same-Side Conjecture

1
3
2
4
5 6
7 8
Let m3 = 82
- m5 = 98
- m6 = ____
- m7 =____
- m8 =____
By the Linear Pair Conjecture

1
3
2
4
5 6
7 8
Let m3 = 82
- m5 = 98
- m6 = 82
- m7 =____
- m8 =____
By the Linear Pair Conjecture

1
3
2
4
5 6
7 8
Let m3 = 82
- m5 = 98
- m6 = 82
- m7 =____
- m8 =____
What is the defined
relationship between 3
and 6?
Alternate Interior Angles!!!

1
3
2
4
5 6
7 8
Let m3 = 82
- m5 = 98
- m6 = 82
- m7 =____
- m8 =____
What is the defined
relationship between 3
and 6?
Alternate Interior Angles!!!

1
3
2
4
5 6
7 8
Let m3 = 82
- m5 = 98
- m6 = 82
- m7 =82
- m8 =____
Which has a corresponding
angle relationship with 3.

1
3
2
4
5 6
7 8
Let m3 = 82
- m5 = 98
- m6 = 82
- m7 = 82
- m8 =____
Which has a corresponding
angle relationship with 3.

1
3
2
4
5 6
7 8
Let m3 = 82
- m5 = 98
- m6 = 82
- m7 = 82
- m8 = 98
Which makes alternate
exterior angle magic with 1.

1
3
2
4
5 6
7 8
Let m3 = 82
- m5 = 98
- m6 = 82
- m7 = 82
- m8 = 98
Now the short
method…

Now the short
method…

If you’re asked to
find, not justify or
prove that angles
are congruent or
have the same
angle measure:
Now the short
method…

If you’re asked to find, not justify or
prove that angles
are congruent or
have the same
angle measure:
All acute angles
are .
Now the short
method…

If you’re asked to - All acute angles
find, not justify or
are .
prove that angles - All obtuse angles
are congruent or
are 
have the same
angle measure:
Now the short
method…

If you’re asked to - All acute angles
find, not justify or
are .
prove that angles - All obtuse angles
are congruent or
are 
have the same
The
sum
of
an
angle measure:
acute and an obtuse
angle = 180
Provided the lines are
parallel.

If you’re asked to - All acute angles
find, not justify or
are .
prove that angles - All obtuse angles
are congruent or
are 
have the same
The
sum
of
an
angle measure:
acute and an obtuse
angle = 180
Formal proof

Given: j || k
Prove: 4  6
4
6
j
3
k
Formal proof

Statement
Reason
4
6
j
3
k
Formal proof

Statement
Reason
m3 + m4 = 180
4
6
j
3
k
Formal proof

Statement
Reason
m3 + m4 = 180 Linear Pair
Conjecture
4
6
j
3
k
Formal proof

Statement
Reason
m3 + m4 = 180 Linear Pair
Conjecture
m3 + m6 = 180
4
6
j
3
k
Formal proof

Statement
Reason
m3 + m4 = 180 Linear Pair
Conjecture
m3 + m6 = 180 Same Side
Interior Angle
Conjecture
4
6
j
3
k
Formal proof

Statement
Reason
m3 + m4 = 180 Linear Pair
Conjecture
m3 + m6 = 180 Same Side
Interior Angle
Conjecture
Transitive
m3 + m4 =
Property
m3 + m6
4
6
j
3
k
Formal proof

Statement
Reason
m3 + m6 = 180 Same Side
Interior Angle
Conjecture
Transitive
m3 + m4 =
Property
m3 + m6
Subtraction
m4 = m6
Property of
Equality
4
6
j
3
k
Formal proof

Statement
m3 + m4 =
m3 + m6
m4 = m6
4  6
Reason
Transitive
Property
Subtraction
Property of
Equality
4
6
j
3
k
Formal proof

Statement
m3 + m4 =
m3 + m6
m4 = m6
4  6
Reason
Transitive
Property
Subtraction
Property of
Equality
Definition of
Congruence
4
6
j
3
k
You will most likely have to do
one of these on your next quiz.

Statement
m3 + m4 =
m3 + m6
m4 = m6
4  6
Reason
Transitive
Property
Subtraction
Property of
Equality
Definition of
Congruence
4
6
j
3
k
If it does ask you to
justify…

If it does ask you to
justify…

Include a
definition or
theorem that
allows you to
state your angle
relationship.
If it does ask you to
justify…

Include a
definition or
theorem that
allows you to
state your angle
relationship.
3.2 Practice # 1
Why is 3 also 132?

Include a
definition or
theorem that
allows you to
state your angle
relationship.
3.2 Practice # 1
Why is 3 also 132?

Include a
definition or
theorem that
allows you to
state your angle
relationship.
3.2 Practice # 1
3:
Vertical Angles
Why is 3 also 132?

Include a
definition or
theorem that
allows you to
state your angle
relationship.
3.2 Practice # 1
3:
Vertical Angles
5:
Why is 3 also 132?

Include a
definition or
theorem that
allows you to
state your angle
relationship.
3.2 Practice # 1
3:
5:
Vertical Angles
Corresponding
Why is 3 also 132?

Include a
definition or
theorem that
allows you to
state your angle
relationship.
3.2 Practice # 1
3:
5:
Vertical Angles
Corresponding
7:
Why is 3 also 132?

Include a
definition or
theorem that
allows you to
state your angle
relationship.
3.2 Practice # 1
3: Vertical Angles
5: Corresponding
7: Alternate Exterior
You try #2

Include a
definition or
theorem that
allows you to
state your angle
relationship.
3.2 Practice # 1
3: Vertical Angles
5: Corresponding
7: Alternate Exterior
Solution

5 is 78 because of
alternate interior
angles.
Solution

5 is 78 because of
alternate interior
angles.
1 is 78 because of
vertical angles.
Be specific!!!

1 is 78 because of
vertical angles.
7 is 78 because of
corresponding
angles
Be specific!!!

1 is 78 because of
vertical angles.
7 is 78 because of
corresponding
angles
Alternative:
7 makes a vertical
angle pair with #5
If you don’t write anything, we
assume you are talking about the
angle measure given to you.

1 is 78 because of
vertical angles.
7 is 78 because of
corresponding
angles
Alternative:
7 makes a vertical
angle pair with #5
Similar idea, moving to
#5

Similar idea, moving to
#5

130 is the reference angle.
Similar idea, moving to
#5

130 is the reference angle.
Angle 1 is _____ because it makes
a __________ ________ with the
130 angle.
Similar idea, moving to
#5

130 is the reference angle.
Angle 1 is 50 because it makes a
linear pair with the 130 angle.
Similar idea, moving to
#5

130 is the reference angle.
Angle 1 is 50 because it makes a
linear pair with the 130 angle.
Angle 2 is
Similar idea, moving to
#5

130 is the reference angle.
Angle 1 is 50 because it makes a
linear pair with the 130 angle.
Angle 2 is 130 because it is a
corresponding angle to the 130.
You do #6

130 is the reference angle.
Angle 1 is 50 because it makes a
linear pair with the 130 angle.
Angle 2 is 130 because it is a
corresponding angle to the 130.
Things to remember in
sketches:

Things to remember in
sketches:

Make sure their exists a
relationship between the angles.
Things to remember in
sketches:

Make sure their exists a
relationship between the angles.
Touch the same transversal, and
that the lines are parallel.
Things to remember in
sketches:

Make sure their exists a
relationship between the angles.
Touch the same transversal, and
that the lines are parallel.
Keep this in mind as we tackle the
remaining problems.
Reversing the process
It seems we’ve gone
down this road before…


Theorem 3-4

Theorem 3-4
If 2 lines and a
transversal form
corresponding angles
that are congruent,
then the lines are
parallel

Theorem 3-5
If 2 lines and a
transversal form
alternate interior
angles that are
congruent, then the
lines are parallel

Theorem 3-6
If 2 lines and a
transversal form same
side interior angles that
are supplementary, then
the lines are parallel

Theorem 3-7
If 2 lines and a
transversal form
alternate exterior angles
that are congruent, then
the lines are parallel
Let’s use the 3.3 Practice to see
how to problem solve…

1–6
Let’s use the 3.3 Practice to see
how to problem solve…

1–6
(A) Find the congruent angles
Let’s use the 3.3 Practice to see
how to problem solve…

1–6
(A) Find the congruent angles
(B) Determine the lines they are
on.
NOT THE TRANSVERSAL!!!

1–6
(A) Find the congruent angles
(B) Determine the lines they are
on.
Let’s use the 3.3 Practice to see
how to problem solve…

1–6
(A) Find the congruent angles
(B) Determine the lines they are
on.
(C) Identify the relationship
between them to justify.
Let’s use the 3.3 Practice to see
how to problem solve…

7:
Poof… A proof…
Let’s use the 3.3 Practice to see
how to problem solve…

8: A walk through…
Let’s use the 3.3 Practice to see
how to problem solve…

9 – 14
Let’s use the 3.3 Practice to see
how to problem solve…

9 – 14
(A) Work under the belief that
the lines are parallel.
Let’s use the 3.3 Practice to see
how to problem solve…

9 – 14
(A) Work under the belief that
the lines are parallel.
(B) Identify the relationship
and set up an equation.
This will be either congruent
or supplementary, if possible.

9 – 14
(A) Work under the belief that
the lines are parallel.
(B) Identify the relationship
and set up an equation.
15 – 20: Some of my
faves…

15 – 20: Some of my
faves…

(1)
Look at the angle pair they provide
you.
15 – 20: Some of my
faves…

(1)
(2)
Look at the angle pair they provide
you.
Identify the relationship, if any,
from the diagram.
15 – 20: Some of my
faves…

(1)
(2)
(3)
Look at the angle pair they provide
you.
Identify the relationship, if any,
from the diagram.
Find the desired value.
15 – 20: Some of my
faves…

#
Relationship
Justification
15 – 20: Some of my
faves…

#
15
Relationship Justification
11 & 10 are
supplementary
15 – 20: Some of my
faves…

#
15
Relationship Justification
11 & 10 are Lines u and t are
supplementary parallel because
same side interior
angles are
supplementary.
15 – 20: Some of my
faves…

#
15
16
Relationship Justification
11 & 10 are Lines u and t are
supplementary parallel because
same side interior
angles are
supplementary.
6  9
15 – 20: Some of my
faves…

#
16
Relationship
6  9
Justification
Lines a and b are
parallel because
alternate interior
angles are
congruent.
You fill in the rest…

#
16
Relationship
6  9
Justification
Lines a and b are
parallel because
alternate interior
angles are
congruent.
You fill in the rest…

#
Relationship
Justification
17
13 and 14
supplementary
Nothing: this is always true no
matter what lines are parallel.
18
13 and 15 are congruent
Lines t and u are parallel
because corresponding angles
are congruent
19
12 is supplementary to 3 3 is also supplementary to 4
because of linear pairs. By the
congruent supplements
theorem, 4 and 12 are congruent,
which are corresponding angles,
making a and b parallel
You fill in the rest…

#
Relationship
Justification
19
12 is supplementary to 3 3 is also supplementary to 4
because of linear pairs. By the
congruent supplements
theorem, 4 and 12 are congruent,
which are corresponding angles,
making a and b parallel
20
2 and 13 are congruent
a and b are parallel since
alternate exterior angles are
congruent.