Chapter 1:
Points, Lines, Planes,
and Angles
Geometry
Mrs. Cotta
Name: __________________
Name:
Period:
Date:
Notes Sec 1.1 and 1.2
Points, Lines, and Planes
By the end of this section, you should be able to:
-
Explain the meaning of equidistant.
-
List the “undefined terms” of geometry and draw/identify them.
-
Use the terms collinear, coplanar, and intersection.
I. Building Blocks of Geometry – The “Undefined Terms”
Point
Line
Plane
Chalkboard Examples:
NOTE: These three terms, (point, line, and plane), are the building blocks of all
geometric figures and are UNDEFINED.
Page | 1
II. More Vocabulary
Term
Definition
Symbol/Example
space
collinear points
noncollinear points
coplanar points
noncoplanar
points
intersection
III. Understanding the Meaning of “Equidistant”
Suppose that you are given the following problem. Read it carefully and determine a
possible solution.
Problem: A grocery store chain would like to open a new store in your town. The
owner would like the grocery store to be 1 mile from his home. He also wants to make
sure that the location he chooses will be the same distance from the nearest bus stop
as it is from the main highway. Using the diagram below, identify the location(s) for the
store.
Owner’s House
Main Highway
Bus Stop
Page | 2
What does equidistant mean?
How many solutions can you find to this problem?
**PROBLEM SOLVING TIP: Always study a diagram and review what you immediately
recognize before looking at the questions.
Using the diagram below, classify statements 1 - 4 as true or false.
1. XY intersects plane M at point O.
X
M
2. T, O, and R are collinear.
S
T
O
R
W
3. R, O, S, and W are coplanar.
Y
4. R, S, T, and X are coplanar.
Use the diagram of the rectangular box to answer questions 5 through 9.
5. Name a fourth point in the same plane as
points A, B, and C.
D
B
A
6. Name a fourth point that is in the same
plane as points D, C, and H.
C
H
E
G
F
7. Are there any points in CG besides C and G?
8. Name the intersection of planes ABFE and BCGF.
9. Name two planes that do not intersect.
Page | 3
Name:
Period:
Date:
Notes Sec 1.3
Segments, Rays, and Distance
By the end of this section, you should be able to:
-
Use symbols for lines, segments, rays and distances.
-
Find distances.
-
State and use the Segment Addition Postulate.
Previous Section Review
1. Name the three undefined “building blocks” of geometry:
2. What does it mean if points are coplanar?
Collinear?
Algebra Review
Find the value of the variable for each equation.
4
a  20
5
1. 7 x  35
2.
4. (2 g  15)  g  9
5. 5 y  3 y 26
3.
e
9
3
6. 2(d  5)  3(d  2)
I. What does “between” mean?
We say a point P is between A and B if ____________________________________
__________________________________________________________________________.
Page | 4
Example of betweenness:
II. Important Vocabulary
Term
Definition
Symbol/Example
Segment AC
Denoted:
Ray AC
Denoted:
Opposite Rays
III. Defining Length
What is the length of the segment between the points 8 and -3?
A method to determine length:
IMPORTANT: When a question asks for the length of a segment, (for example, the length of
AC ), the length is denoted by: _____________________________________.
Page | 5
IV. Postulates
What is a postulate?
How does a postulate differ from a theorem?
Postulate 2
Segment Addition
Postulate
If B is between A and C, then
**PROBLEM SOLVING TIP: Always sketch a diagram of the problem before trying to
solve it.
Example: Suppose B is between A and C, with AB = x, and BC = x + 6, and AC = 24.
a) Find the value of x.
b) Find BC.
IV. Midpoints, Bisectors, and Congruent Segments
Term
Definition
Symbol/Example
Congruent
Congruent
segments
Midpoint of
Segment
Bisector of Segment
Page | 6
Examples for Section 1.3
1. Does the symbol represent a segment, line, ray or length?
a) AC
b) AC
2. Is AB the same as BA?
c) AC
d) AC
How about AC and CA?
Or AC and CA?
3. Suppose -2 and 11 are the coordinates of two points on a number line. What is the
distance between them?
4. Imagine a number line. Is it possible to list all numbers that fall between 1 and 2 on
the number line?
5. E is the midpoint of AC . If AE = 5x + 3 and EC = 33, find the value of x.
6. Given points G, E, and H with E between G and H. Use the information given and
determine GE, EH, and the value of x. Is E the midpoint?
GE = x + 2
GH = 20
EH = 2x – 6
7. Using the number line shown, name the graph of the given inequalities.
a) x  4
A B C D E F G H I J
-4 -2 0 2 4 6 8 10 12 14
b) x  2
Page | 7
Name:
Period:
Date:
Notes Sec 1.4
Angles
By the end of this section, you should be able to:
-
Name an angle.
-
Find the measure of an angle.
-
State and use the Angle Addition Postulate.
Previous Section Review
1. Using the points A and B, write the notation for a line, ray, segment, and the
distance from A to B.
2. State the segment addition postulate:
3. Classify each statement as true or false.
a. All points on a line are coplanar.
b. A line has one endpoint.
c. A point is named by a capital letter.
d. Two lines intersect in two points.
e. The edge of a plane is a line.
F. Points have no size.
4. Using the number line shown, name the graph of the given inequalities.
a) x  4
A B C D E F G H I J
-4 -2 0 2 4 6 8 10 12 14
b) x  2
Page | 8
I. Angle Components
Angle:
Sides:
Vertex:
Naming Angles:
II. Classifying Angles
Term
Definition
Example
Acute Angle
Right Angle
Obtuse Angle
Straight Angle
III. Using Your Protractor
2
1
Page | 9
III. Postulates
If point B lies in the interior of AOC, then
______________________________________________________.
Postulate 4
Angle Addition
Postulate
If AOC is a straight angle and B is any point not
on AC, then
_______________________________________________________.
IV. More Vocabulary
Term
Definition
Example
Congruent Angles
Adjacent Angles
Bisector of an Angle
Page | 10
Examples for Section 1.4
Name the vertex and sides of the given angle.
C
4
1. 4
3
2. 1
E
7
B
2
6
5
D
1
A
3. 6
4. Name all angles adjacent to 6.
5. How many angles can you name with D as the vertex?
6. m7 + m6 = m __________
7. m2 + m3 = __________
Given the diagram below, state whether you can reach the conclusion given.
C
8. mCOF = 50
9. mAOC = 90
10. mDOC = 180
11. AO = OB
F
A
O
50
B
E
D
12. AOC  BOC
13. mDOF = 130
14. Points E, O, and F are collinear.
15. Point C is in the interior of AOF.
16. AOE and AOD are adjacent angles.
17. AOB is a straight angle.
Page | 11
Name:
Period:
Date:
Notes Sec 1.5
Postulates and Theorems Relating Points, Lines, and Planes
By the end of this section, you should be able to:
-
Use postulates and theorems relating points, lines, and planes.
Previous Section Review
1. Find the measures of 2, 3, and 4 when the measure of
1 is:
a) 90
2 1
3 4
b) 93
c) t
I. Postulates
Postulate
Postulate 5
Postulate 6
Postulate 7
Diagram
A line contains at least two points; a
plane contains at least three points not
all in one line; space contains at least
four points not all in one plane.
Through any two points there is exactly
one line.
Through any three points there is at least
one plane, and through any three noncollinear points there is exactly one
plane.
Page | 12
Postulate 8
If two points are in a plane, then the line
that contains the points is in that plane.
Postulate 9
If two planes intersect, then their
intersection is a line.
“exactly one” = ____________________________
What is the purpose of postulates?
II. Theorems
Theorems are ____________________________________________________________.
Theorem
Diagram
Theorem 1-1
If two lines intersect, then they intersect
in exactly one point.
Theorem 1-2
Through a line and a point not in the
line there is exactly one plane.
Theorem 1-3
If two lines intersect, then exactly one
plane contains the lines.
Page | 13
Examples for Section 1.5
1. Try to draw a diagram that shows two lines intersecting in more than one point.
State the postulate that makes this situation impossible.
2. Do two intersecting lines determine a plane?
3. Do three points determine a plane?
4. Look at the intersection of the ceiling and the front wall of the classroom and let this
be line l. Choose a point C on the classroom floor.
a. Is there a plane that contains l and C?
b. State the theorem that applies.
5. Reword Theorem 1-3 as two statements, one describing existence and one
describing uniqueness.
Page | 14
Geometry ~ Chapter 1 ~ Important Vocabulary, Postulates, & Theorems
Note: You should be able to define, explain, and draw an example of each of the
following terms. Review your notes! For additional help, come see me or go to math
lab!
Point
Line
Plane
Space
Collinear Points
Noncollinear Points
Coplanar Points
Noncoplanar Points
Intersection
“Between”
Segment
Ray
Opposite Rays
Congruent
Congruent Segments
Midpoint of Segment
Bisector of Segment
Postulate
Segment Addition Postulate
Angle Addition Postulate
Congruent Angles
Adjacent Angles
Bisector of an Angle
Postulate 5
Postulate 6
Postulate 7
Postulate 8
Postulate 9
Theorems 1-1 through 1-3
Page | 15