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Chapter 1:
1-1:
(page 1)
Points, Lines, Planes, and Angles
A Game and Some Geometry
(page 1)
In the figure below, you see five points: A,B,C,D, and E. Use a centimeter ruler to find the requested
distances. Please measure to the nearest tenth of a centimeter.
C
A
B
D
E
1. The distance from A to C is
cm and the distance from B to C is
cm.
2. The distance from A to D is
cm and the distance from B to D is
cm.
3. Both points C and D are said to be
they are equally distant from A and B.
from A and B because
4. Another point on the diagram that is equidistant from A and B is point
.
5. The number of points that are equidistant from points A and B are
.
6. The geometric figure that would be formed by all these points is a
.
Locate a point 9 cm from point A and call it point F. Locate all the points that are 9 cm from point A.
7. The number of points that are 9 cm from point A is
.
8. The geometric figure that would be formed 9 cm from point A is a _______________.
Locate a point that is 9 cm from points A & B and call it point G. Locate all the points that are 9 cm from
points A & B.
9. The number of points that are 9 cm from points A and B is ___________________.
10. The geometric figure that would be formed by connecting these points is a
Assignment: Written Exercises, pages 3 & 4: 1 to 10
.
1-2:
Points, Lines, and Planes
(page 5)
What real life objects are made up of points (dots)?
Look up the definitions for Point, Line, and Plane in the glossary of your textbook.
POINT:
LINE:
PLANE:
Point, Line, and Plane are the basis to defining other geometry terms, they are accepted without
, therefore they are considered
terms.
Descriptions of the Three (3) Undefined Terms
picture
symbol
description
-------------------------------------------------------------------------------------------------------------------POINT
* has
length, width, or thickness
* has
dimensions
* occupies
space
-------------------------------------------------------------------------------------------------------------------LINE
* has
, but
width or thickness
* has an
set of points that
extends in
directions
-------------------------------------------------------------------------------------------------------------------PLANE
* has
and
but
thickness
* has an
set of points that
extends in
*
directions
surface
SPACE: the set of
points.
GEOMETRIC FIGURE: a
of points.
[figure 1]
A
[figure 2]
B
V
C
W
X
Y
D
Z
COLLINEAR POINTS: points that lie on the same
.
example from figure 1:
NONCOLLINEAR POINTS: points that do
lie on the same line.
example from figure 1:
COPLANAR POINTS: points that lie in the same
.
example from figure 2:
NONCOPLANAR POINTS: points that do
example from figure 2:
lie in the same plane.
INTERSECTION (of two or more figures): the set of
to all figures.
examples:
(1) Lines
(2) Planes
Assignment: Written Exercises, pages 7 to 9: 1 to 35 odd #’s
Prepare for Quiz on Lessons 1-1 & 1-2
that are common
Segments, Rays, and Distance
1-3:
Point B is
(page 11)
A and C if it lies on line AC (
A
B
).
C
SEGMENT: consists of points A and C and all points that are
example:
A
A and C.
C
RAY: consists of AC and all other points P such that C is
example:
A
C
A and P.
P
OPPOSITE RAYS: two rays that have the same
example:
X
O
and form a line.
Y
Geometry and Algebra are brought together with the
A
|
-4
B
|
-3
C
|
-2
D
|
-1
Every point is paired with a
E
|
0
line.
F
|
1
G
|
2
H
|
3
I
|
4
J
|
5
. Every number is paired with a
.
1 - to - 1 Correspondence
A corresponds to
or A ! _______,
B corresponds to
or B ! _______ , etc.
LENGTH (of a segment): the
AB means the
between its endpoints.
of AB or the
between points A & B.
AB = a - b = b - a
examples: CH = ______
AJ = ______
BF = ______
POSTULATES (Axioms): statement accepted without
POSTULATE 1
.
RULER POSTULATE
(1)
The points on a line can be paired with the real numbers in such a way that any two points can
have coordinates
and
. (coordinatized line)
(2)
Once a coordinate system has been chosen in this way, the distance between any two points equals
the absolute value of the difference of their
. (distance)
POSTULATE 2
SEGMENT ADDITION POSTULATE
If B is between A and C, then:
AB + BC =
.
example: Given the diagram with AB = 2x, BC = x-1, and AC = 23, find AB and BC.
A
B
C
AB =
BC =
CONGRUENT: objects that have the same _____________ and _____________.
CONGRUENT SEGMENTS: segments that have equal _____________.
example:
A
C
AB
CD
! AB ____ CD
B
D
_______ expresses a relationship between numbers.
_______ expresses a relationship between geometric figures, NOT numbers.
MIDPOINT of a SEGMENT: the point that divides the segment into_______ congruent
segments.
example:
A
M
B
AM ____ MB, then AM ____ MB
! M is the _____________ of AB.
BISECTOR of a SEGMENT: a _____________, _____________, _____________, or
plane that intersects the segment at its midpoint.
Assignment: Written Exercises, pages 15 & 16: 1 to 45 odd #’s and 46, 47
Angles
(page 17)
ANGLE: a figure formed by 2 rays that have the same
.
1-4:
example:
Naming an angle may be done as follows:
.
To measure angles (in degrees in this course), use a
.
Classification of Angles
Acute Angle: measure is between
and
Right Angle: measure equals
degrees.
degrees.
Obtuse Angle: measure is between
and
Straight Angle: measure equals
degrees.
degrees.
Angle in a Plane (exclude straight angle)
Plane is separated into 3 parts:
(1) the angle itself
(2) the interior of the angle
(3) the exterior of the angle
How many angles are shown in the diagram below?
Name the different angles:
X
2
3
O
Y
Z
PROTRACTOR POSTULATE
POSTULATE 3
On line AB in a given plane, choose any point between A and B. Consider ray OA and ray OB and all rays
that can be drawn from O on one side of line AB. These rays can be paired with the real numbers from 0 to
180 in such a way that:
(a) ray OA is paired with
, and ray OB with
.
(b) If ray OP is paired with x, and ray OQ with y, then
Q
A
=
x!y
.
P
O
B
ANGLE ADDITION POSTULATE
POSTULATE 4
If point B lies in the interior of !AOC , then:
+
=
A
.
B
O
B
C
A
O
C
If !AOC is a straight angle and B is any point not on line AC, then:
+
=
180º .
CONGRUENT ANGLES: angles that have
measure.
example:
m!A _____ m!B
A
B
!"A _____ "B
ADJACENT ANGLES: two angles in a
that have a common vertex and a
common side, but no common interior points. (adj. !' s)
examples: Are ! 1 and ! 2 adjacent angles? Circle Yes or No.
(1)
(2)
(3)
1
1
2
1
Yes or No
2
2
Yes or No
(4)
(5)
Yes or No
(6)
1
2
2
1
1 2
Yes or No
Yes or No
Yes or No
BISECTOR of an ANGLE: the ray that divides the angle into 2 congruent
m!XOY _____ m!YOZ
X
example:
then !XOY _____ !YOZ
O
! OY __________ "XOZ
Z
angles.
MORE EXAMPLES: In diagram, OB bisects !AOC.
A
2
3
D
(1)
(2)
B
1
O
m! 1 = 2x + 5 and m! 2 = 3x -12.
C
m! 1 = ________
m! 2 = ________
m! 3 = ________
m! 1 = x + 4 and m! 3 = 2x + 8.
m! 1 = ________
m! 2 = ________
m! 3 = ________
Assignment: Written Exercises, pages 21 & 22: 1 to 35 odd #’s, 24
Prepare for Quiz on Lessons 1-3 & 1-4
1-5:
Postulates and Theorems Relating Points, Lines, & Planes
The phrases, “exactly one” and “one and only one” imples
POSTULATE 5
A line contains at least
A plane contains at least
Space contains at least
and
.
points;
points not all in one line;
points not all in one plane.
POSTULATE 6
Through any two points there is exactly
POSTULATE 7
Through any three points there is at least
line.
plane;
Through any three noncollinear points there is exactly
plane.
POSTULATE 8
If
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
THEOREMS: statements that can be
.
.
NOTE: Writing proofs will be covered in Chapter 2.
THEOREM 1-1
If two lines intersect, then they intersect in exactly
point.
NO PROOF - example:
THEOREM 1-2
Through a line and a point not in the line there is exactly
plane.
NO PROOF - example:
THEOREM 1-3
If two lines intersect, then exactly
NO PROOF - example:
(page 22)
plane contains the lines.
Relationships between points:
Two points
be collinear.
Three points
be collinear or noncollinear.
Three points
be coplanar.
Three noncollinear points determine a
Four points
.
be coplanar or noncoplanar.
Four noncoplanar points determine
Space contains at least
.
noncoplanar points.
Three ways to determine a plane:
(1) _________________________ noncollinear points determine a plane,
ex:
(2) A line and a point not on the
determine a plane,
ex:
(3) _________________________ intersecting lines determine a plane,
ex:
Relationships between two lines in the same plane:
Two lines are either parallel or they
in exactly one point.
examples:
Relationships between a line and a plane:
A line and a plane are either parallel, or they
the plane
the line.
in exactly one point, or
examples:
Relationships between two planes:
Two planes are either parallel or they
in a line.
examples:
Assignment: Written Exercises, pages 25 & 26: 1 to 11 odd #’s, 13 to 17 ALL, & 19
Prepare for Test on Chapter 1: Points, Lines, and Planes