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Warm-up
• Open book to page 73
– Read all of the 3.1 section
– Some questions to think about…
•
•
•
•
What is the difference between parallel and skew?
Can segments and rays be parallel?
What can 2 planes be? Parallel? And?
What can a line and a plane be?
Chapter 3
Parallel Lines and Planes
• Learn about parallel
line relationships
• Prove lines parallel
• Describe angle
relationship in
polygons
3.1 Definitions
Objectives
• State the definition of parallel lines
• Describe a transversal
Parallel Lines ( || )
Coplanar lines that do not intersect.
m || n
m
n
Parallel Lines (
or
)
The way that we mark that two lines are
parallel is by putting arrows on the lines.
m || n
m
n
Skew Lines ( no symbol  )
Non-coplanar lines, that do not intersect.
q
p
What is the difference between the
definition of parallel and skew lines?
Parallel Planes
Planes that do not intersect.
P
Q
Can a plane and a line be parallel?
1.
2.
3.
4.
5.
Parallel, intersecting , or skew?
Name parallel lines
Name skew lines
Names 5 lines parallel to plane ABCD
Name parallel planes
Theorem
2 levels of a parking structure are parallel
• They need to be reinforced by support beams (AB and CD)
• What do you think you can say about the beams based on what you
know about the 2 levels (planes)?
B
P
A
If two parallel planes are cut by a
third plane, the the lines of
intersection are parallel.
Q
C
D
The Transversal
t
r
s
Any line that
intersects two or
more coplanar
lines in different
points.
Name the transversal
• If j, k and l are coplanar name the
transversal.
l
j
k
Name the transversal
• If j, k and l are coplanar name the
transversal.
k
j
l
Name the transversal
• If j, k and l are coplanar name the
transversal.
j
k
l
Special Angle Pairs
exterior
t
1 2
3
interior
5
7
4
6
8
exterior
r
s
Understanding the
position of these
special angles is
key!!
When dealing with
parallel lines, the
relationships become
more specific
Corresponding Angles
(corr. s)
t
Think
1 2
3
4
r
- shape
5
7
6
8
s
Corresponding Angles
t
1 2
3
5
7
4
6
8
r
s
1 and  5
3 and  7
2 and  6
4 and  8
Alternate Interior Angles
(alt. int. s)
• Think
t
1 2
3
4
r
- shape
5
7
6
8
s
Alternate Interior Angles
t
 4 and  5
 3 and  6
1 2
3
5
7
4
6
8
r
s
Same Side Interior Angles
(s-s. int s)
• Think
t
1 2
3
4
r
- shape
5
7
6
8
s
Same Side Interior Angles
 4 and  6
 3 and  5
t
1 2
3
5
7
4
6
8
r
s
**TRANSVERSAL**
• What is unique about the
transversal in relation to
any of the special angle
r
pairs?
t
1 2
3
5
7
4
6
8
s
• The transversal line
will always be a part of
making up BOTH
ANGLES.
Group Practice
• Name the two lines and the transversal that form
each pair of angles.
– Re-draw the diagram with only the lines you need
– Label the “exterior” and “Interior” areas of the diagram
1
5
k
2
6
9 10
13 14
l
4
3
7
11 12
15 16
t
8
n
**THE TRANSVERSAL IS ALWAYS THE LINE THAT HELPS
CREATE THE PAIR OF ANGLES**
 1 and  3
k
1 2
5 6
9 10
13 14
l
4
3
7
11 12
15 16
t
8
n
 3 and  11
k
1 2
5 6
9 10
13 14
l
4
3
7
11 12
15 16
8
t
n
 10 and  11
k
1 2
5 6
9 10
13 14
l
4
3
7
11 12
15 16
t
8
n
 6 and  9
k
1 2
5 6
9 10
13 14
l
4
3
7
11 12
15 16
t
8
n
 8 and  11
k
1 2
5 6
9 10
13 14
l
4
3
7
11 12
15 16
t
8
n
Remote Time
True or False
• A transversal intersects only parallel lines.
True or False
• Skew lines are not coplanar
True or False
• If two lines are coplanar, then they are
parallel.
True or False
• If two lines are parallel, then exactly one
plane contains them.
(A) Alternate interior angles
(B) Same-side interior angles
(C) Corresponding angles
1
2
5
3
6
7
A
 2 and  7
4
8
B
D
E
C
F
G
H
I
(A) Alternate interior angles
(B) Same-side interior angles  ADE and
 DEB
(C) Corresponding angles
1
2
5
3
6
7
A
4
8
B
D
E
C
F
G
H
I
(A) Alternate interior angles
(B) Same-side interior angles  BEF and
 EGI
(C) Corresponding angles
1
2
5
3
6
7
A
4
8
B
D
E
C
F
G
H
I
3.2 Properties of Parallel Lines
Objectives
• Learn the special angle relationships
…when lines
are parallel
Lesson Focus
• “IF”
- what we know
• ‘Then” – what we can prove
• In this lesson we know that we have 2
parallel lines being cut by a transversal
If 2 || lines are CBT, then….
1. Corresponding angles are congruent
– corr. Ls are 
2. Alternate interior angles are congruent
– alt. int. Ls are 
3. Same-side interior angles are supplementary
– S-S int. Ls are Supp.
IF YOU ONLY REMEMBER 1
THING….
• When 2 parallel lines
are CBT the whole
diagram will only
have 2 different angle
measurements!!!
Name all angles that are congruent
to  1
1
2
3 4
5
6
7 8
Name all angles that are
supplementary to  1
1
2
3 4
5
6
7 8
If m  5 = 60, then m  4 =_____
and m  2 = _____
1
2
3 4
5
6
7 8
If m  7 = 110, then m  3 =_____
and m  4 = _____
1
2
3 4
5
6
7 8
Group work
• Complete the proofs for alt. interior angles
Postulate
If two parallel lines are cut by a
transversal, then corresponding angles are
congruent.
t
If 2 || lines are CBT, the
corr. Ls are congruent
r
1 2
3 4
Can you name the
corresponding angles?
5 6
7
8
s
Theorem
If two parallel lines are cut by a
transversal, then alternate interior angles
t
are congruent.
r
1 2
If 2 || lines are CBT, then
alt. int. Ls are congruent
Can you name the alt.
int. angles?
4
3
5
7
6
8
s
Theorem
If two parallel lines are cut by a
transversal, then same side interior angles
t
are supplementary.
r
1 2
If 2 || lines are CBT, then
s-s int. Ls are supp.
Can you name the s-s
int angles?
4
3
5
7
6
8
s
Theorem
A line perpendicular to one of two
parallel lines is perpendicular to the
other.
t
r
*Based on the
position of these
angles what did we
already learn in this
lesson that told us this
was true?
s
Group Practice
• Find the values of x and y
80º
2yº
xº
Group Practice
• Find the values of x and y
xº
3yº
Group Practice
• Find the values of x and y
xº
yº
50º
60º
Group Practice
• Find the values of x and y
110º
xº
2yº
WARM-UP
• S-S Int Angle Proof
3.3 Proving Lines Parallel
Objectives
• Learn about ways to prove lines are parallel
• Use Theorems about parallel lines
• Define an auxiliary line
• How many of the red lines can you
place going through the point that
would be parallel the black line?
• How many red lines can you show
as perpendicular?
In your notes
• Black and white postulate sheet
• Answer the following
– What is the difference between post. 10 and 11?
– What is my evidence in each?
– What do I prove in each one?
• 3.2 – The parallel lines told us about the
special angle relationships.
• 3.3 – Special angle relationship help us
determine that the lines are parallel.
TELL ME WHY THE LINES ARE
PARALLEL?
“The lines are parallel because the
________________________________________.”
FILL IN THE BLANK WITH ANY OF THE
FOLLOWING….
1. Corresponding angles are congruent
2. Alternate interior angles are congruent
3. Same side interior angles are supplementary
4. 2 lines are perpendicular to the same line
5. 2 lines are parallel to the same line
Postulate
If two lines are cut by a transversal and the
corresponding angles are congruent, then
the lines are parallel.
If 2 lines are CBT and the
corr. Ls are congruent,
then the lines are ||.
1
m
2
n
If 1  2, then
m || n.
Theorem
If two lines are cut by a transversal and the
alternate interior angles are congruent,
then the lines are parallel.
If 2 lines are CBT and the
alt. int. Ls are congruent,
then the lines are ||.
2
m
1
n
If 1  2, then
m || n.
Theorem
If two lines are cut by a transversal and the
same side interior angles are supplementary,
then the lines are parallel.
If 2 lines are CBT and the
s-s int. Ls are supp, then
the lines are ||.
m
1
2
If 1 suppl  2, then
n
m || n.
Theorem
In a plane two lines perpendicular to the
same line are parallel.
t
If t  m and t  n , then
m || n.
m
n
Theorem
Two lines parallel to the same line are
parallel to each other
If p  m and m  n, then
p
p  n
m
n
**THINK
TRANSITIVE
PROPERTY
Partner Work (skip)
• Complete proof of theorem 3-5
Experiment - Goal
• Find parallel lines
Materials
• 5 Pencils
• Paper
• Which pair of lines are parallel?
– “Line ___ and line ____ are parallel because
the _______________________________.”
• Which pair of lines are parallel?
– “Line ___ and line ____ are parallel because
the _______________________________.”
l
t
8
Al&t
6
j
Bj&k
C  none
1
2
7
3
4
5
k
m6=m4
l
t
8
Al&t
6
j
Bj&k
C  none
1
2
7
3
4
5
k
m  2 + m  3= m  5
l
t
8
Al&t
6
j
Bj&k
C  none
1
2
7
3
4
5
k
m  2 + m  3 + m  8 = 180
l
t
8
Al&t
6
j
Bj&k
C  none
1
2
7
3
4
5
k
71
l
t
8
Al&t
6
j
Bj&k
C  none
1
2
7
3
4
5
k
m  1 =m  8 = 75
l
t
8
Al&t
6
j
Bj&k
C  none
1
2
7
3
4
5
k
 5 and  6 are supplementary
l
t
8
Al&t
6
j
Bj&k
C  none
1
2
7
3
4
5
k
 4 and  5 are supplementary
l
t
8
Al&t
6
j
Bj&k
C  none
1
2
7
3
4
5
k
 2 and  3 are complementary and
m  5 = 90
l
t
8
Al&t
6
j
Bj&k
C  none
1
2
7
3
4
5
k
TELL ME WHY THE LINES ARE
PARALLEL?
“The lines are parallel because the
________________________________________.”
FILL IN THE BLANK WITH ANY OF THE
FOLLOWING….
1. Corresponding angles are congruent
2. Alternate interior angles are congruent
3. Same side interior angles are supplementary
4. 2 lines are perpendicular to the same line
5. 2 lines are parallel to the same line
Ch. 3 QUIZ
• Definition of parallel and skew lines
• When two parallel lines are cut by a transversal
– Name different types of pairs of angles
– Be able to name all the angles congruent to a certain
angle
– Find measurements of angles (Go off the information
given, not assumptions of the diagram.)
• Question directly from 3.3 worksheet
• Know the 5 ways to prove lines parallel
3.4 Angles of a Triangle
Objectives
• Classify Triangles
• State the Triangle
Sum Theorem
• Apply the Exterior
Angle Theorem
Triangle
• A triangle is a figure formed by three
segments
– Joining noncollinear points.
Not in a straight
line
Draw a
triangle in
your notes
Vertex
Each of the three points is a vertex of the
triangle. (plural vertices)
A
C
▲ABC
B
Sides
• The segments are the sides of the triangle
A
AB
AC
BC
C
B
Types of Triangles
(by sides)
• WHAT DO YOU THINK ARE WAYS WE
CAN CLASSIFY (NAME) TRIANGLES
IN TERMS OF THEIR SIDES?
Isosceles
2
congruent
sides
Equilateral
Scalene
All sides
congruent
No congruent
sides
Types of Triangles
(by angles)
Can you name
these triangles?
Equiangular
Acute
Right
Obtuse
3 acute
angles
1 right angle
1 obtuse
angle
Experiment 1
• GOAL: Find the sum of the
angles of a triangle
• Write down the goal of the experiment,
answer the questions in your notes as we go
along
Materials
• Paper Triangle
• Pencil
Procedure
1. Label the angles of the triangle  1,
 2 and  3
2. Draw a point at the tip of each
vertex
3. Rip off the three angles of the triangle
1
2
3
4. Put the three angles in a row so that
the angles meet at one point and at
least one side of each angle touches
the side of another angle
5. What is the sum of those angles? 
Hint use the Angle Addition
Postulate.
1
1
3
2
Something else to think about
Can you draw a triangle where the sum of the
angles is not 180 degrees?
Theorem
The sum of the measures of the angles of a
triangle is 180
B
mA + mB + mC = 180
C
A
**This is true for any and all triangles!!!
Remote Time
A – Acute
B – Obtuse
C - Right
• State whether a triangle with two angles
having the given measures is acute, obtuse,
or right.
• 55
• 43
A – Acute
B – Obtuse
C - Right
• 47
• 43
A – Acute
B – Obtuse
C - Right
Corollaries (pg.94)
• Corollaries are just like theorems, but are so
closely related to a single theorem, that they
are listed as being associated with that
theorem.
• Just like theorems, they can be used as
reasons in proofs
• THESE COROLLARIES ARE BASED
OFF A TRIANGLE = 180.
Corollary
1. If two angles of one triangle are congruent
to two angles of another triangle, then the
third angles are _________.
Corollary
1. If two angles of one triangle are congruent
to two angles of another triangle, then the
third angles are congruent.
• Why?
Corollary
2. Each angle of an equiangular triangle has
measure___.
Corollary
2. Each angle of an equiangular triangle has
measure 60º.
• Why?
Corollary
3. In a triangle, there can be at most one
_____ or _______ angle.
Corollary
3. In a triangle, there can be at most one right
or obtuse angle.
• Why?
http://www.mathsisfun.com/triangle.html
Corollary
4. The acute angles of a right triangle are
_____________.
Corollary
4. The acute angles of a right triangle are
complimentary.
• Why?
• Find the measure of  A
80º
50º
A
50º
Experiment 2
• GOAL: Explore the exterior
angles of a triangle
Materials
•
•
•
•
Protractor
Pencil
Paper
Ruler
Copy the diagram
 1 is an exterior angle
One triangle has 6 different exterior angles
m1 + m 2 = 180
 3 and  4 are called
remote interior angles
 2 is the adjacent interior angle
Measure the four angles of your triangle using
the protractor. Be precise !
 m1=
 m 2=
 m3=
 m4=
•
Do you notice any measurements that
are equal? Any that add up to 180?
Theorem
The measure of an exterior angle of a triangle
equals the sum of the measures of the two
remote interior angles
m2 +
m1
m2 + m3 + m4
= 180
= 180
m1 = m3 + m4
• Use what you know about parallel lines,
triangles, and ext angles to solve….
3.5 Angles of a Polygon
Objectives
• Define a regular and
convex polygon
• Calculate the
interior and exterior
angles of a polygon
Polygon
Means “many angles”
The Polygon
1. Each segment intersects exactly two other
segments, one at each endpoint.
2. No two segments with a common endpoint
are collinear
**Your shape must enclose an area with its sides,
therefore each side is connected, with no overlaps**
Not Polygons
Can you explain why each of these figures is
NOT a polygon ?
Not Polygons
Not a segment
Not Polygons
Can you explain why each of these figures is
NOT a polygon ?
Not Polygons
Doesn’t intersect 2 other segment one at each
endpoint
Not Polygons
Can you explain why each of these figures is
NOT a polygon ?
Not Polygons
Exactly 2 other segment
Convex Polygon
No line containing a side of the polygon
contains a point in the interior of the
polygon.
convex
Non convex
The Diagonal
A segment that joins non-consecutive
vertices.
Choose one
“start”
point
What polygon
does not have
any diagonals?
# of
sides
Picture
Name
Names of Polygons
Number of sides
3
4
5
6
8
10
n
Name
Triangle
Quadrilateral
Pentagon
Hexagon
Octagon
Decagon
n-gon
# of
sides
Picture
Name
# of
triangles
1. Choose one
point on the
polygon and
draw all the
possible
diagonals
from that
point.
2. How many
triangles do
you see ?
# of
sides
Picture
Name
# of
Interior
triangles angle
sum
1. One triangle
has an angle
sum of 180.
2. What is the
angle sum
inside the
polygon?
(n-2)180
• How did we come up with this formula?
• Look at your tables
– What do you notice about the number of sides
compared to the number of triangles that can be
drawn from the diagonals?
4 sides
2 triangles
=(4-2)180
=(2) 180
5 sides
3 triangles
=(5-2)180
=(3) 180
Regular Polygon (p.103)
 All angles congruent
 All sides congruent
http://www.mathsisfun.com/geometry/polygons-interactive.html
REGULAR POLYGON
# of
sides
Picture
Name
# of
Interior
triangles angle
sum
One
interior
angle
One
exterior
angle
Exterior
angle
sum
Exterior Angle + Interior Angle =
Backside of worksheet
Theorem
The sum of the measures of the exterior angles,
one at each vertex, of a convex polygon is 360.
4
1
1
3
3
2
1 + 2 + 3 = 360
2
1 + 2 + 3 + 4 = 360
**THIS APPLIES TO ANY CONVEX POLYGON!!
REGULAR POLYGONS
• All the interior angles are congruent
• All of the exterior angles are congruent
360
n
= the measure of
each exterior angle
**The only time you can divide by n is when you have a
regular poly.
Remote Time
• A – Always
• B – Sometimes
• C – Never
A – Always
B – Sometimes
C – Never
• The sum of the measures of the exterior
angles of any polygon one angle at each
vertex is ___________ 360º
A – Always
B – Sometimes
C – Never
• The sum of the measures of the interior
angles of a convex polygon is
____________ 360º
A – Always
B – Sometimes
C – Never
• The sum of the measures of the exterior
angles of a polygon __________ depends of
the number of sides of a polygon.
White Board
•
Find the interior angle sum
1. Heptagon
– Find the measure of one interior and
exterior angle
1. Regular Decagon
White Board
• An exterior angle of a regular polygon has
measure 10. The polygon has _____ sides.
• 36 sides
White Board
• An interior angle of a regular polygon has
measure 160. The polygon has _____ sides.
• 18 sides
White Board
• Three of the angles of a quadrilateral have
measures 90, 60, and 115. The fourth angle
has measure ___ .
• 95
3.6 Inductive Reasoning
Objectives
• Discover the uses and hidden dangers of
inductive logic
• Compare inductive and deductive logic
Inductive Logic
Archimedes
• The conclusion is
based on seeing a
pattern from
observations.
• Is probably true, but
need not be.
• Can be dangerous
because of its tendency
to allow generalization
from specific
information.
Deductive Logic
• Based on accepted
statements
–
–
–
–
Definitions
Postulates
Theorems
Etc…
• Conclusion must be
true.
• Two-column proofs
are deductive.
Pythagoras
Remote time
• A – Inductive Reasoning
• B – Deductive Reasoning
A – Inductive Reasoning
B – Deductive Reasoning
• Mary has given Jimmy a present on each of
his birthdays. He reasons that she will give
him a present on his next birthday.
A – Inductive Reasoning
B – Deductive Reasoning
• John knows that multiplying a number by -1
changes the sign of the number. He reasons
that multiplying a number by an even power
of -1 will change the sign of the number an
even number of times. He concludes that
this is equivalent to multiplying a number
by +1.
A – Inductive Reasoning
B – Deductive Reasoning
• Ramon noticed that sloppy joes had been on
the school menu the past 5 Fridays. Ramon
decides that the school always serves sloppy
joes on Fridays.
White Board
• Look for a pattern and predict the next two
numbers in each sequence.
• 1, 1, 2, 3, 5, ___, ___
• 1, 1, 2, 3, 5, 8, 13
• 1,3,5,___,___,
• 1, 3, 5, 7, 9
• 12, 7, 2, ___, ___
• 12, 7, 2, -3, -8
Ch. 3 test
• Study and understand all of the postulates in
the chapter – you will have to fill in the
blanks
• Definition of parallel and skew lines
• Understanding the formula used for the
interior and exterior angles of a polygon
o How can you use it to find the measurement of
one interior angle of a regular polygon
– Use it to find how many sides a polygon is
based off the exterior angle measurement
• Given a diagram – solve for the given angles and
variables
o **NEED TO BE ABLE TO DISTINGUISH WHICH
TWO LINES ARE PARALLEL AND WHICH LINE
IS TRANSVERSAL – HOW IS EACH ANGLE
BEING FORMED?
o Other things to help solve angles and variables..
–
–
–
–
–
Is a bigger angle made up of two smaller ones??
Vertical angles
Supplementary angles
The sum of a triangle is 180
Exterior angle = 2 remote interior angles
• Using inductive reasoning to solve a number
pattern
• Being able to prove which lines parallel based off
of certain angles being congruent
• How many lines can be parallel and perpendicular
to a point outside a line
• Write the postulate or theorem word for word
– Hint: it will be the ones form 3.2 or 3.3
• Fill in and writing an entire proof