Download Points, Lines & Planes

Points, Lines, Planes and Angles
Points, Lines and Planes
These basic concepts of geometry are theoretical and cannot be
precisely defined. They do not exist in the physical world.
Concept
A point is a location in
space.
Dimension
Representation
It has no dimensions,
no length, no width,
and no thickness.
Points A, B, and M
A line is a set of
connected points. It is
assumed to be straight
and is determined by
two specific points.
It has one dimension,
an infinite length, but
no width or thickness.
Line a and line PQ
A plane is an infinite
large flat surface.
It has two dimensions,
an infinite length and
infinite width, but no
thickness.
B
A
M
a
P
Plane
Q
A line segment is a portion of a line between two points called endpoints. A
line segment is named using its endpoints.
Line segment AB
A
B
A ray is half a line and its endpoint. It is a part of a line that has only one
endpoint and goes on forever in one direction. A ray is named by using
the endpoint and some other point on the ray. The endpoint is always
mentioned first.
Ray PQ
P
Q
Some Basic Concepts of Angles
An angle is the union of two rays with a common endpoint. The rays are
called sides and the common point is called the vertex. An angle can be
named in different ways:  Oor AOBor BOA
A
O
B
P is said to be the interior of the  AOB whereas Q is exterior to the angle.
A
Q
P
O
B
Forming an angle by rotation:
Given OA, OB on the same line and extended in the same direction.
If OA is rotated counterclockwise about O, we say that BOA is
an angle of rotation generated by OA.
A
te rmina l
O
OB is called the initial side and
OA is called the terminal side.
initia l
B
When the terminal side rotates until it falls on the initial side,
We say that it has made one complete revolution.
Equal angles are angles formed by the same amount of rotation, and they
are also called congruent angles.
Angles formed by a counterclockwise rotation are called positive and
those formed by a clockwise rotation are called negative angles.
A
A
pos itive
ne ga tive
B
O
B
Units of measurement:
- degree – 1/360 of a complete revolution
- minute – 1/60 of a degree
- second – 1/60 of a minute
The notation for the measure of angle  O is m O
Classification of Angles
Acute Angle
has a measure between 0 and 90.
Right Angle
has a measure of 90 . A right angle is
angle formed by a quarter of a revolution.
Obtuse Angle
has a measure between 90 and 180.
Straight Angle
has a measure of 180. A straight angle is an
angle formed by one half of a revolution.
The sides of a straight angle extended in
opposite direction, forming a straight line.
Reflex Angle
has a measure between 180 and 360.
Adjacent Angles
two angles with same vertex and
a common side between them.
2
1
Complementary Angles
two angles whose some is a right angle.
2
1
Supplementary Angles
two angles whose some is a straight angle.
Vertical Angles
two nonadjacent angles, each less than a
straight angle, formed by the same measure.
Vertical angles have the same measure.
Angles a,b and c,d are vertical angles.
2
1
c
b
a
d
Names of Angles Pairs Formed by a Transversal
Intersecting Parallel Lines
Parallel lines never meet.
The distance between them is always the same.
l
p
l p
Intersecting lines cross at exactly one point.
P
a
b
Perpendicular lines are intersecting lines that
form right angle.
a
a b
A line that intersects two other lines at different points is called transversal.
Names of Angles Pairs Formed By Transversal Intersecting Parallel Lines
Name
Description
Alternate interior
angles
Interior angles that do
not have a common
vertex on alternate
sides of transversal
Sketch
t
1
4
8
Alternate exterior
angles
5
7
Exterior angles that do
not have a common
vertex on alternate
sides of transversal
2
l
3
6
p
t
1
4
8
Corresponding
angles
Angle Pair
Described
5
7
One interior and one
exterior angle on the
same side of the
transversal
2
l
3
6
p
t
1
4
8
5
7
6
2
l
3
p
 4and 6
 3and 5
 1and 7
 2and 8
 1and 5
 2and 6
 3and 7
 4and 8
Property
Alternate interior
angles have the
same measure.
m 4  m 6
m 3  m 5
Alternate exterior
angles have the
same measure.
m 1  m 7
m 2  m 8
Corresponding
angles have the
same measure.
m 1  m 5
m 2  m 6
m 3  m 7
m 4  m 8
Exercises
• Identify each pair of complementary
angles.
15
C
D
65
75
 AOBand BOC
 CODand DOE
B
25
O
E
A
• Identify each pair of supplementary angles.
A
140
B
40
D
Q
140
40
C
 AQBand BQC
 CQDand DQE
Exercises
• Identify the angles that are congruent
and their measure .
F
A
E
 ABCand BCD
m ABC  m BCD  42
C
42
42
B
 ABFand ECD
m ABF  m ECD  138
D
• Find the measure of all angles in each figure.
112
1
1
63
?
?
13
72
2
2
3
27
167
m 1  m 3  108
m 2  72
6
4
7
3
5
m 1  m 2  m 5  m 6  68
m 3  m 4  m 7  112
Exercises
• Given m  A=(3x+15) and m  B=(2x-5) find x if
• a) the two angles are complementary
3x+15+2x-5=90
x=18
• b) the two angles supplementary
3x+15+2x-5=180
x=34
Self Check
Identify each pair of complementary
angles.
14
A
C
D
150
B
Q
30
150
C
20
E
O
B
30
70
76
Identify each pair of supplementary
angles.
A
Given m A=100 and m B=(3x-7).
Find the measure of 1 if line l is
parallel to line p.
Find x if A and  B are
supplementary angles.
l
1
110
p
40