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Lesson 1-1
Point, Line, Plane
Modified by Lisa Palen
1-1 Part A - Geometry :
The Objects
Point, Line, Plane,
Segment, Ray,
Angle
Undefined Terms
•Point
•Line
•Plane
We describe these, rather than defining them.
Point
• A place in space. Has no actual size.
• How to Sketch:
Use dots
• How to label:
Use capital printed letters
Never name two points with the same letter
(in the same sketch).
A
B
C
A
Line
•
Straight figure, extends forever, has no thickness or width.
•
How to Sketch:
•
Use arrows at both ends
How to label:
(1) Use small script letters – line
•
A
n
(2) Use any two points on the line Never name a line using three points.
Never name two points with the same letter
(in the same sketch).
B
C
n
Plane
• Flat surface that extends forever in all directions.
• How to sketch: Use a parallelogram (four sided figure)
• How to name: 2 ways:
(1) Use a capital script letter – Plane M
(2) Use any 3 noncollinear points in the plane
M
A
Vertical Plane
ACB
BAC
BCA
CAB
B
CBA
C
Horizontal Plane
ABC
Other
More Objects
•Segment
•Ray
•Angle
Segment
Definition: part of a line that includes two points (called the
endpoints) and all points between them
A
B
How to sketch:
How to name:
AB or BA
The symbol AB is read as "segment AB".
AB (without a symbol) means the length of
the segment or the distance between points
A and B.
Ray
Definition: Part of a line starting at one point (called the endpoint)
And extending forever in one direction.
R
C
How to sketch:
D
How to name:
( the symbol RA is read as “ray RA” )
A
Y
RA or RY ( not RAY )
What is
?
Angle
Definition: Angle - Figure formed by two rays with a common
endpoint, called the vertex. The two rays are called
sides of the angle
ray
vertex
ray
Naming an angle: (1) Using 3 points
(2) Using 1 point
(3) Using a number – next slide
Using 3 points: vertex must be the middle letter
This angle can be named as ABC or CBA
Using 1 point: using only vertex letter
* Use this method is permitted when the vertex point is the vertex
of one and only one angle.
Since B is the vertex of only this angle, this can
also be called B .
B
A
C
Naming an Angle - continued
Using a number: A number (without a degree symbol) may be
used as the label or name of the angle. This
A
number is placed in the interior of the angle near
its vertex. The angle to the left can be named
B
2
C
as 2 .
* The “1 letter” name is unacceptable when …
more than one angle has the same vertex point. In this case, use
the three letter name or a number if it is present.
Example

K is the vertex of more than one angle.
Therefore, there is NO K in this diagram.
There are LKM , PKM , and LKP .
There is also 2 and 3 but there is no 5!!!
L
M
2
K
3
P
Lesson 1-1 Part B
Vocabulary
Collinear, Coplanar,
Intersection, Intersect,
Parallel, Perpendicular
Modified by Lisa Palen
Collinear Points
Definition: Collinear points are points that lie
on the same line. (The line does not
have to be visible.)
A B
B
C
Points A, B and C
are collinear.
A
C
Points A, B and C
are noncollinear
Coplanar
Definition: Coplanar objects (points or lines) are
objects that lie on the same plane.
(The plane does not have to be visible.)
S
A
Q
D
B
R
P
P, Q, R and S are coplanar.
C
A, B, C and D are noncoplanar.
Coplanar
Definition: Coplanar objects (points or lines) are
objects that lie on the same plane.
(The plane does not have to be visible.)
A
D
B
C
E
H
F
G
Are they coplanar?
ABC ?
yes
ABCF ?
NO
HGFE ? yes
DCEF ?
yes
AGF ?
yes
CBFH ?
NO
Intersection / Intersect
• Definition The intersection of two objects is the set of points
in common to both objects. (where the objects touch.)
r
I
g
• Definition Two objects intersect if they have pointsr
in
common. (if the objects touch.)
g r
The intersection of line
Line
and line
g
and line
intersect at point I.
is point I.
Intersection of Two Lines
skew
• If two lines intersect, what is their
intersection?
Intersection
is a point.
parallel
• Otherwise, they are either parallel or skew.
Parallel Lines
• Two coplanar lines that don’t intersect
• Symbol: ║ means “is parallel to”
v
v║w
Parallel lines
go in
the same
direction.
w
Perpendicular Lines
• lines that intersect at right angles
m
• Illustration:
mn
n
• Symbol:  means “is perpendicular to”
• Key Fact: 4 right angles are formed.
Lesson 1-2
Segments and Rays
Modified by Lisa Palen
Recall: What is a Segment?
Definition: two points (called the endpoints) and all points
between them
A
B
How to sketch:
How to name:
AB or BA
The symbol AB is read as "segment AB".
Measure (of a Segment)
Definition: The length of the segment or the distance between
the two endpoints
A
The measure of
Recall: The symbol
“segment A B”.
Notation:
B
is AB.
is read as
AB (without a symbol) means the length of
the segment or the distance between points
A and B.
Congruent Segments
B
Definition: Congruent segments are segments
with equal measures (lengths).
A
Mark congruent segments with . . dashes..
C
Congruent segments have the same number of dashes.
Notation: The symbol
congruent to”.

means “is
AB  CD EF  GH
D
E
F
G
H
AB  EF
Congruent Segments
B
Using the Notation:
A
C
Numbers are equal. Objects are congruent.
AB: the distance from A to B ( a number )
AB: the segment AB ( an object )
AB = CD
AB  CD
Incorrect notation: AB  CD
AB = CD
Correct notation:
D
Midpoint
Definition: A midpoint is a point that divides a
segment into two congruent segments.
E is the midpoint of
and
.
D
DE = EF
E
F
Segment Bisector
Definition: A segment bisector is ANY object that divides
a segment into two congruent segments.
A
F
A
B
E
AB bisects DF.
B
D
E
AB bisects DF.
D
F
A
Plane M bisects DF.
E
D
F
AB bisects DF.
B
Postulates
Definition: a statement we accept as true without proof.
Examples:
• Through any two points there is
exactly one line.
• Through any three non-collinear points,
there is exactly one plane.
Postulates
Examples:
• If two lines intersect,
then the intersection is a point.
• If two planes intersect,
then the intersection is a line.
The Ruler Postulate
The points on any line can be paired with the real numbers
in such a way that:
The Ruler Postulate says
youpoints
can use
a paired
ruler with
to 0 and 1.
• Any two chosen
can be
• The distancemeasure
between any
twodistance
points in a number line is
the
the absolute value of the difference of the real numbers
between
any
two
points!
corresponding to the points.
(It also gives us a
.
formula.)
The Ruler Postulate
• So, we can measure the distance between two points
using a “ruler”.
•Formula: take the absolute value of the difference of the
two coordinates a and b: │a – b │
|
|
PK = 3 - -2 = 5
(distance is always positive)
Reminder
• The coordinates are the numbers on the ruler or
number line!
• The capital letters are the names of the points.
G
H
I
J
K
L
M
N
O
P
-5
• Coordinates: -3, -2, -1, 0, 1, 2, 3, etc.
• Points: G, H, I, J, etc.
Q
R
5
S
Another Example
Find the distance between I and S.
G
H
I
J
K
L
-5
M
N
O
P
Q
R
S
5
Coordinate of I: -4
Coordinate of S: 6
Take the absolute value of the difference: │a – b │
│ -4 - 6 │= │ - 10 │ = 10
Finding the Midpoint
(of Two Points on a Number Line)
The coordinate of a midpoint of a segment whose
endpoints have coordinates a and b is
ab
2
G
H
-5
I
J
K
L
M
N
O
P
Q
R
5
S
Example
Find the coordinate of the midpoint of the segment PK.
G
H
I
J
K
L
M
N
O
P
Q
-5
R
5
a  b 3  (2) 1

  0.5
2
2
2
Now find the midpoint on the number line.
S
So what do we mean by between?
Which picture shows, “C is between A and B”?
So “C is between A and B” means that C is ON the
segment AB .
Okay, but this is
not the
definition. 
Between
or The Segment Addition Postulate
Definition:
•If C is between A and B, then AC + CB = AB.
•If AC + CB = AB, then C is between A and B.
AC + CB = AB
between
AC + CB > AB
not between
This is also called the Segment Addition Postulate.
The Segment Addition Postulate
(This is the same as “between.” )
In Other
Words:
Or:
The whole is the sum of the parts.
Part + Part = Whole
These are the same length.
The Segment Addition Postulate
Example: If C is between A and B, AC = 4 and CB = 8,
then find AB.
Step 1: Draw.
B
8
Step 2: Label.
A 4 C
Step 3: Find equation.
(Substitute)
Step 4: Solve.
Step 5: Make sure you
answer the question.
Part + Part = Whole
AB
AC + CB = AB
4 + 8 = AB
12 = AB
The Segment Addition Postulate
Example: If E is between D and F, DE = 5 and DF = 15,
then find EF.
Step 1: Draw.
Step 2: Label.
Step 3: Find equation.
(Substitute)
Step 4: Solve.
Step 5: Make sure you
answer the question.
Part + Part = Whole
EF
5
15
DE + EF = DF
5 + EF = 15
EF = 10
Midpoint
Example:
If E is the midpoint of
then find EF and DF.
Step 1: Draw.
Step 2: Label.
Step 3: Find equation.
(Substitute)
Step 4: Solve.
Step 5: Answer question.
DE = EF
5 = EF
D
5
, and DE = 5,
E
5
DE + EF = DF
Part = Part
Part + Part = Whole
5 + 5 = DF
10 = DF
F
Lesson 1-4
Angles
Angle
Definition: Angle - Figure formed by two rays with a common
endpoint, called the vertex. The two rays are called
sides of the angle
ray
vertex
ray
Angles and Points
• Angles can have points
in the interior, in the
exterior, or on the
angle.
E
A
D
B
Points A, B and C are on the angle, D is in the
interior and E is in the exterior.
n B is the vertex.
n
C
Interior / Exterior of an Angle
Definition (you don’t need to memorize this.)
A point is in the interior of an angle if it does not lie on the
angle itself and it lies on a segment whose endpoints are on
the sides of the angle.
A, B, and C are on the angle.
An exterior point is a point that is
neither on the angle nor in the interior
of the angle.
A
Interior
Point
E
D
Exterior
Point
B
C
The Protractor Postulate
You don’t need to memorize this!
Given aThe
ray AB
and a
Protractor
Postulate
number r between
0 and
180,
says you
can
use a
there is exactly
one raytowith
protractor
measure
endpoint A extending to
angles!
either side of AB, such that
the measure of the angle
formed is r degrees.
The Ruler and Protractor
Postulates
The Ruler Postulate lets us use a ruler to measure the distance between two
points.
The Protractor Postulate lets us use a protractor to measure an angle..
Protractor Applet
Measuring Angles
Just as we can measure segments, we can also
measure angles.
We use units called degrees to measure angles.
– A circle measures _____
360º
?
?
180º
– A half-circle measures _____
90º
?
– A quarter-circle measures _____
– One degree is the angle measure of 1/360th
of a circle.
Measure (of an Angle)
Definition: The size of the angle
A
B
Notation:
The measure of ABC is
mABC
Angles are measured using units
called degrees (in this class.)
C
4 Types of Angles
Acute Angle: an angle whose measure is less than 90.
A
Right Angle: an angle whose measure is exactly 90.
B
Obtuse Angle: an angle whose measure is greater than
90 and less than 180.
Straight Angle: an angle that measures exactly
C
180 .
D
Lesson 1-4: Angles
56
Congruent Angles
Definition: Congruent angles - angles that have equal measures
Congruent angles are marked with the same number of “arcs”.
The symbol for congruence is 
Example:
2   4.
4
2
Lesson 1-4: Angles
57
Angle Bisector / Bisect
An angle bisector is a ray that splits the angle into
two congruent angles. The ray bisects the angle.
Example 1: Since 3   5, BD bisects ABC .
A
41° K
B
41°
j
4
6
5
3
D
C
U
Example 2: UK is an angle bisector.
Lesson 1-4: Angles
58
Example 1 Angle Bisector
If ML is an angle bisector of PMY and
m PML = 68, then find:
• m  PMY = _______
• m  LMY = _______
Example 2 Angle Bisector
If ML is an angle bisector of  PMY and
m PMY = 86, then find:
• m  PML = _______
• m  LMY = _______
Adding Angles
When you want to add angle measures, use the
notation m1, meaning the measure of  1.
If you add m1 + m2, what is your result?
mADC = 36 + 22
A
B
mADC = 58
36°
22°
1
D
C
2
How did you know to add???
Angle Addition Postulate
That last example is an example of
The Angle Addition Postulate:
If D is in the interior of ABC,
DBC = m< _____
ABC
then m< ABD
____ + m< ____
If mABD + mDBC = m ABC,
then D is in the interior of ABC.
Angle Addition Postulate
A simpler way to remember this postulate:
whole
part
part
part + _______
part = _________
whole
_______
Lesson 1-5
Pairs of
Angles
Lesson 1-5: Pairs of Angles
65
Adjacent Angles
Definition: A pair of coplanar angles with a common (shared)
vertex and common side that do not have overlapping
interiors.
Examples: 1 and 2 are adjacent. 3 and 4 are not.
1 and ADC are not adjacent.
A
B
4
36°
22°
1
C
3
2
D
Adjacent Angles( a common side )
Lesson 1-5: Pairs of Angles
Non-Adjacent Angles
66
Complementary Angles
Definition: A pair of angles whose sum of measures is 90˚
m2 = 50°
Examples:
A
B
2
Q
A
B
F
2
1
C
Q
 1 and  2 are
adjacent
complementary angles.
( have a common side )
m1 = 40°
1
R
G
 1 and  2 are
complementary but not
adjacent angles.
have a common side 67)
Lesson 1-5: Pairs(ofdon’t
Angles
Supplementary Angles
Definition: A pair of angles whose sum of measures is 180˚
Examples:
m1 = 40°
m2 = 140°
B
1 and  2 are adjacent
supplementary angles.
2
Q
A
1 and  2 are
supplementary but not adjacent
angles.
2
A
C
B
F
1
Q
R
Lesson 1-5: Pairs of Angles
1
G
68
Opposite Rays
X
A
Y
opposite rays
not opposite rays
D
E
DE and ED are not opposite rays.
Opposite Rays
Definition:
Two rays with the same endpoint, that together form a line.
Or (better): Two rays with the same endpoint that together
form a straight angle.
X
A
Y
AX and AY are opposite rays.
XAY is a straight angle
Linear Pair
Definition: A linear pair is a pair of adjacent angles
whose non-adjacent rays form opposite
rays.
2
A
m1 = 40°
m2 = 140°
Q
B
1
C
1 and  2 are a linear pair.
Another
A linear pair is a pair of adjacent
“Definition”: supplementary angles.
Lesson 1-5: Pairs of Angles
71
Vertical Angles
Definition: A pair of non-adjacent angles formed by
intersecting lines.
Examples:
1 and 3
2 and 4
Another Definition: A pair of angles whose sides form opposite
rays.
The pairs of opposite rays are QA & QC and QB & QD
Lesson 1-5: Pairs of Angles
72
Postulates vs. Theorems
Definition: A postulate is a statement we accept as true
without proof.
Examples: Segment Addition Postulate and
Angle Addition Postulate
Definition: A theorem is a statement we use logic to
show is true.
Examples: Linear Pair Theorem and
Vertical Angles Theorem (next slides)
Theorem (Linear Pairs)
A linear pair is supplementary.
Given:
Prove:
B
1 and 2 are a linear pair.
1 and 2 are supplementary.
A
Statements
1. 1 & 2 are linear pair.
2. QAand QC are opposite rays.
3. AQC is a straight angle.
4. mAQC = 180
5. m1 + m2 = mAQC
6. m1 + m2 = 180
7. 1 and 2 are supplementary.
2
Q
Reasons
1
C
1. Given
2. Defn. linear pair
3. Defn. opposite rays
4. Defn. straight angle
5. Angle Addition Postulate
6. Substitution Property
7. Defn. supplementary
Lesson 1-5: Pairs of Angles
Vertical Angles Theorem
Theorem
Vertical angles are congruent.
Lesson 1-5: Pairs of Angles
75
Theorem: Vertical angles are
congruent.
A
Given:
Prove:
The diagram
Statements
1. 1 & 2 and 2 & 3 are linear pairs
2. 1 & 2 and 2 & 3 are suppl.
3. m1 + m2 = 180,
m2 + m3 = 180
4. m1 + m2 = m2 + m3
5. m1 = m3
6. 1  3
1
4
1  3
D
B
Q
2
3
Reasons
C
1. Defn. linear pair/diagram
2. Linear pairs are
supplementary.
3. Defn. supplementary
4. Substitution Property
5. Subtraction Property
6. Defn. Congruent Angles
What’s “Important” in Geometry?
4 things to always look for !
180˚
360˚
. . . and
Congruence
90˚
Most of the rules (theorems)
and vocabulary of Geometry
are based on these 4 things.
Lesson 1-5: Pairs of Angles
77
Algebra and Geometry
Common Algebraic Equations used in Geometry:
(
(
(
(
)+(
)+(
)+(
)=( )
)=( )
) = 90˚
) = 180˚
If the problem you’re working on has a variable (x),
then consider using one of these equations.
Lesson 1-5: Pairs of Angles
78
Example: If m4 = 67º, find the
measures of all other angles.
Step 1: Mark the figure with given info.
Step 2: Write an equation.
m3  m4  180
67º
m3  67 180
3
4
2
1
m3 180  67  113
Because 4 and 2 arevertical angles, they are equal. m4  m2  67
Because 3 and 1 are vertical angles, they are equal. m3  m1  117
Lesson 1-5: Pairs of Angles
79
Example: If m1 = 23 º and m2 = 32 º, find the
measures of all other angles.
Answers:
m4  23 (1 & 4 are vertical angles.)
m5  32 (2 & 5 are vertical angles.)
m1  m2  m3  180
2
23  32  m3  180
m3  180  55  125
1
m3  m6  125
3 & 6 are vertical angles.
Lesson 1-5: Pairs of Angles
3
6
4
5
80
Example: If m1 = 44º, m7 = 65º find the
measures of all other angles.
Answers: m3  90
m1  m4  44
m4  m5  90
44  m5  90
m5  46
4
5
6
3
2
1
7
m6  m7  90
m6  65  90
m6  25
Lesson 1-5: Pairs of Angles
81