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Angles, lines, rays, and segments
MATHEMATICS
Contents
• Recap the terms
• Test Yourself - 1
• Angles in daily life
• Congruent angles
• What is an angle?
• Naming an angle
• Pairs of angles: Types
• Test Yourself - 2
• Interior and exterior of an angle
• Pairs of angles formed
by a transversal
• Measurement of angle
• Test Yourself - 3
• Types of angle: Right angle
Obtuse angle
Acute angle
Straight angle
Recap Geometrical Terms
Point
An exact location on a plane is
called a point.
Line
A straight path on a plane,
extending in both directions
with no endpoints, is called a
line.
Line
segment
A part of a line that has two
endpoints and thus has a
definite length is called a line
segment.
Ray
A line segment extended
indefinitely in one direction
is called a ray.
Angles In Daily Life
If we look around us, we will see angles everywhere.
What Is An Angle ?
When two non-collinear rays join with a common endpoint
(origin) an angle is formed.
Ray BA
A
B
Common
endpoint
B
C
Ray BC
Common endpoint is called the vertex of the angle. B is the
vertex of ABC.
Ray BA and BC are two non-collinear rays
Ray BA and ray BC are called the arms of ABC.
Fact: We can also think of an angle formed by rotating one
ray away from its initial position.
Naming An Angle
To name an angle, we name any point on one ray, then the
vertex, and then any point on the other ray.
A
B
C
For example: ABC or CBA
We may also name this angle only by the single letter of the
vertex, for example B.
Interior And Exterior Of An Angle
An angle divides the points on the plane into three regions:
• Points lying on the angle
(An angle)
A
P
X
• Points within the angle
(Its interior portion. )
R
F
B
T
C
• Points outside the angle
(Its exterior portion. )
Measurement Of An Angle
Protractor is used to measure and draw angles.
Angles are accurately measured in degrees.
Types Of Angles
There are four main types of angles.
Right angle
Acute angle
A
B
Obtuse angle
A
A
C
B
B
C
Straight angle
A
B
C
C
Right angle: An angle whose measure is 90 degrees.
Straight Angle
Right Angle
Acute Angle
Obtuse Angle
Examples Of Right Angle
Obtuse angle: An angle whose measure is greater than 90
degrees.
Straight Angle
Right Angle
Acute Angle
Obtuse Angle
Examples Of Obtuse Angle
Acute angle: An angle whose measure is less than 90
degrees.
Straight Angle
Right Angle
Acute Angle
Obtuse Angle
Examples Of Acute Angle
Straight angle: An angle whose measure is 180 degrees.
Straight Angle
Right Angle
Acute Angle
Obtuse Angle
Examples Of Straight Angle
Which of the angles below is a right angle, less than a right
angle and greater than a right angle?
1.
2.
D
E
P
R
Q
F
Greater than a right angle
3.
Right angle
B
A
C
Less than a right angle
Congruent Angles
Two angles that have the same measure are called congruent
angles.
D
A
B
300
C
E
300
F
Congruent angles have the same size and shape.
Pairs Of Angles : Types
• Adjacent angles
• Vertically opposite angles
• Complimentary angles
• Supplementary angles
• Linear pairs of angles
Adjacent Angles
Two angles that have a common vertex and a common ray
are called adjacent angles.
A
A
C
D
Common ray
B
D
B
Common
vertex
E
C
F
Adjacent
ABD
and DBC
ABC
and Angles
DEF arenot
adjacent
angles
Adjacent angles do not overlap each other.
Vertically Opposite Angles
Vertically opposite angles are pairs of angles formed by two
lines intersecting at a point.
C
A
P
B
D
APC = BPD
APB = CPD
Four angles are formed at the point of intersection.
Vertically opposite angles are congruent.
Point of intersection ‘P’ is the common vertex of the four
angles.
Complimentary Angles
If the sum of two angles is 900, then they are called
complimentary angles.
A
D
600
B
E
C
300
F
ABC and DEF are complimentary because
ABC + DEF
600 + 300 = 900
Contd….
If the sum of two angles is more than 900 or less than 900,
then they not complimentary angles.
D
p
700
E
Q
F
300
R
DEF and PQR are not complimentary because
DEF + PQR
700 + 300 = 1000
Supplementary Angles
If the sum of two angles is 1800 then they are called
supplementary angles.
A
P
1000
Q
800
R
B
C
PQR and ABC are supplementary, because
PQR + ABC
1000 + 800 = 1800
Contd….
If the sum of two angles is more than 1800 or less than 1800,
then they are not supplementary angles.
D
A
1100
B
800
C
E
F
DEF and PQR are not supplementary because
ABC + DEF
1100 + 800 = 1900
Linear Pair Of Angles
Two adjacent supplementary angles are called linear pair
of angles.
A
600
C
1200
P
APC + APD
600 + 1200 = 1800
D
Name the adjacent angles and linear pair of angles in the
given figure:
Adjacent angles:
ABD and DBC
A
ABE and DBA
00
30
30
Linear pair of angles:
EBA, ABC
EBD, DBC
0 0
9090
E
D
0
60
600
B
C
Name the vertically opposite angles and adjacent angles in
the given figure:
C
A
P
B
D
Vertically opposite angles: APC and BPD
Adjacent angles: APC and CPD
APB
and CPD
APB
and BPD
Pairs Of Angles Formed by a Transversal
A line that intersects two or more lines at different points is
called a transversal.
G
A
C
Line L (transversal)
P
B
Line M
Line N
Q
D
F
FourLine
angles
are formed
atMpoint
Pparallel
and
four
point
Line
M and
line
N
areline
lines.P
L intersects
line
and
N another
at point
andatQ.
Eight angles are formed in all by the transversal L.
Q by the transversal L.
Pairs Of Angles Formed by a Transversal
• Corresponding angles
• Alternate angles
• Interior angles
Corresponding Angles
When two parallel lines are cut by a transversal, pairs of
corresponding angles are formed.
L
Line L
GPB = PQE
G
A
D
P
B
Q
E
F
Line M
Line N
GPA = PQD
BPQ = EQF
APQ = DQF
Four pairs of corresponding angles are formed.
Corresponding pairs of angles are congruent.
Alternate Angles
Alternate angles are formed on opposite sides of the
transversal and at different intersecting points.
L
G
A
D
Line L
P
B
Q
E
Line M
Line N
BPQ = DQP
APQ = EQP
F
Two pairs of alternate angles are formed.
Pairs of alternate angles are congruent.
Interior Angles
The angles that lie in the area between the two parallel lines
that are cut by a transversal, are called interior angles.
L
G
A 120
Line L
0P
600
E
Line N
APQ + DQP = 1800
1200
600
D
B
Line M
BPQ + EQP = 1800
Q
F
interior
eachside
pairofadd
AThe
pairmeasures
of interiorofangles
lieangles
on the in
same
theup to 1800.
transversal.
Name the pairs of the following angles formed by a
transversal.
GG
G
AA
500
Line
Line
Line LL
L
P P
BBB
1300
D
DD
Q
Q
Q
EEE
FFF
Line
Line
Line
MM M
Line
Line
N
Line
NN