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BRIDGE COURSE
VIKASANA
GEOMETRY-BASICS
Vikasana - CET 2012
Vikasana - CET 2012
An exact position or location
Vikasana - CET 2012
You are here.
In fact, you cannot see a
true point.
P
Point P
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A series of points that go
endlessly in both directions.
Because a line has no width or height,
you cannot see a true line.
A
B
Line AB
Ray is the part of a line that has one
endpoint and goes on endlessly
in the other direction.
C
Ray CD
D
A part of a line that has
two endpoints.
E
F
Line Segment EF
INTERSECTING LINES: Lines that cross
PARALLEL LINES:
Lines that never cross and are
always the same distance apart
Perpendicular Lines
Two lines that intersect to form four right angles
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Ray
Parallel Lines
Intersecting Lines
Line Segment
X
Y
Z
X
Y
∠XYZ
Z
90 °
R
∠USR =
90
∠RST=90
c
S
U
∠USV =
90
T
∠TSV = 90
V
UT
⊥
RV
R
c
U
S
V
T
45°° acute
135°°
obtuse
180 ° is a straight angle
Parallel lines have the same slope
or steepness.
These 2 lines are not parallel, but
they are not intersecting either.
These lines are called skew lines.
S
K
E
W
E
R
intersecting
parallel
perpendicular
90º
skew
The figure formed when two rays share the
same endpoint
Right Angle:
An angle that forms a
square corner
Acute Angle:
Obtuse Angle:
An angle less
than a right angle
An angle greater
than a right angle
Angles In Daily Life
If we look around us, we will see angles
everywhere.
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 BCVikasana
are called
the
arms of ∠ABC.
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Z
Naming An Angle
To name an angle, we name any point on one ray,
then the vertex, and then any point on the other
A
ray.
B
C
For example: ∠ABC or ∠CBA
We may also name this angle only by the single
letter of the vertex,Vikasana
for example
∠B.
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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
• Points within the
P
angle
(Its
interior
X
R
portion. )
• Points outside the
F
B
angle
(Its
exterior
C
T
portion. )
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Measurement Of An Angle
Protractor is used to measure and draw angles.
Angles are accurately measured in degrees.
70
The unit used to
measure angles.
Types Of Angles
There are four main types of angles.
A
A Right angle
B
A
Acute angle
B
C
Obtuse angle
B
C
Straight angle
A
B
C
C
Acute angle: An angle whose measure is less than 90
degree.
Straight Angle
Right Angle
Obtuse Angle
Acute Angle
Examples Of
Acute Angle
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Right angle: An angle whose measure is 90 degree.
Straight Angle
Right Angle
Acute Angle
Obtuse Angle
Examples Of
Right Angle
Obtuse angle: An angle whose measure is greater than 90
degree and less than 180 degree.
Straight Angle
Right Angle
Acute Angle
Obtuse Angle
Examples Of
Obtuse Angle
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Straight angle: An angle whose measure is 180
degree.
Straight Angle
Right Angle
Acute Angle
Obtuse Angle
Examples Of Straight Angle
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Which of the angles below is a right angle,
less than a right angle 2. P
and greater than a right angle? Q
1.
R
Greater than a right angle
D
3.
E
F
Right angle
B
A
C
Less than a right angle
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Pairs Of Angles : Types
•Congruent angles
• Adjacent angles
• Vertically opposite angles
• Complimentary angles
• Supplementary angles
•Linear pairs of angles
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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.
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Adjacent AnglesAdjacent angles are angles
which have a common
side and a common
vertex but no
interior points in common.
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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
E
vertex
C
F
Adjacent Angles ∠ABD and ∠DBC
∠ABC and ∠DEF are not adjacent angles
Adjacent angles do not overlap each other.
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Vertically Opposite Angles- Vertically Opposite
Angles are the angles opposite to each other when
two lines cross.
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Vertically Opposite Angles
Vertically opposite A
angles are pairs of
angles formed by two
lines intersecting at
a point.
B
C
∠APC = ∠BPD
P
D
∠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.
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Complimentary Angles
If the sum of two angles is 900, then they are called
complimentary angles.
A
D
600
B
E
300
F
C
∠ABC and ∠DEF are complimentary because
∠ABC + ∠DEF
600 + 300 = 900
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D
If the sum of two angles
is more than 900 or less than 900,
then they are not
complimentary angles. p
700
E
Q
F
300
R
∠DEF and ∠PQR are not complimentary because
∠DEF + ∠PQR
700 + 300 = 1000
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Supplementary Angles
P
If the sum of two angles
is 1800 then they are
A
Called supplementary
angles.
0
100
800
B
R
C
∠PQR and ∠ABC are supplementary, because
Q
∠PQR + ∠ABC
1000 + 800 = 1800
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If the sum of two angles
is more than 1800 or less than
0, then they are not
180
D
A
supplementary angles.
1100
B
800
C
E
F
∠DEF and ∠PQR are not supplementary because
∠ABC + ∠DEF
1100 + 800 = 1900
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Linear Pair Of Angles
Two adjacent supplementary angles are called
linear pair of angles.
A
600
C
1200
P
∠APC + ∠APD
600 + 1200 = 1800
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D
Name the adjacent angles and linear pair of angles
in the given figure:
A
Adjacent angles:
0
30
300
∠ABD and ∠DBC
∠ABE and ∠DBA
Linear pair of angles:
900900
E
∠EBA, ∠ABC
D
0
60
600
B
C
∠EBD, ∠DBC
Name the
C
A
vertically opposite
angles and adjacent
angles in the given
B
P
D
figure:
Vertically opposite angles: ∠APC and ∠BPD
Adjacent angles: ∠APC and ∠CPD
APB∠and
∠CPD
∠APB∠and
BPD
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Pairs Of Angles Formed by a Transversal
A line that
G Line L (transversal)
intersects two or
P
more lines at A
B
different points
Q
C
D
is called a transversal.
Line M
Line N
F
Four angles
formed
point
another
four
pointQ.Q by the
Line M at
and
line
Nand
are
parallel
lines.at
Lineare
L intersects
line
MP
and
line
N at point
P and
Eight angles are formed in all by the transversal L.
transversal L.
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Pairs Of Angles Formed by a
Transversal
• Corresponding angles
• Alternate angles
• Interior angles
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Corresponding Angles- When two lines are
crossed by another line the angles in matching
corners are called corresponding angles.
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Corresponding Angles
When two parallel lines are cut by a transversal,
L
pairs of corresponding angles are formed.
Line L G
A
D
P
B
Q
E
F
Line M
Line N
∠GPB = ∠PQE
∠GPA = ∠PQD
∠BPQ = ∠EQF
∠APQ = ∠DQF
Four pairs of corresponding angles are formed.
Corresponding pairs of angles are congruent.
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Alternate Angles
Alternate angles are formed on opposite sides of the
L
transversal and at different intersecting points.
Line L
G
A
D
P
B
Q
E
Line M
∠BPQ = ∠DQP
Line N
∠APQ = ∠EQP
F
Two pairs of alternate angles are formed.
Pairs of alternate angles are congruent.
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Interior Angles
Line L
G
0
120
The angles that lie in the area between the two
parallel lines that are cut by a transversal
P
are called interior angles.
600
A
B
600
D
1200
E
Q
F
Line M
Line
LineNN
∠BPQ + ∠EQP = 1800
∠APQ + ∠DQP = 1800
A pair of interior angles lie on the same side of the transversal.
The measures of interior angles in each pair add up to 1800.
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Name the pairs of the following angles
Line
Line
Line LL
L formed by a transversal.
G
G
G
AA
500
P P
BBB
Line
Line
Line
MM M
1300
D
DD
Q
Q
Q
EEE
Line
Line
N
Line
NN
FFF
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More Angles
Reflex Angle- An angle that is greater then
180° and less than 360° is known as reflex
angle.
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Alternate Interior Angles- When two lines are
crossed by another line the pairs of angles on
opposite sides of the transversal but inside the
two lines are called
Alternate Interior
Angles.
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Alternate Exterior Angles- When two lines are
crossed by another line the pairs of angles on
opposite sides of the transversal but outside
the two lines are
called Alternate
Exterior Angles.
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A plane
is a flat
surface that has
length & width
but no height.
A true plane goes on forever in all
directions.
A true plane goes on forever in all
directions.
A true plane goes on forever in all
directions.
A true plane goes on forever in all
directions.
Planes can
intersect.
Planes can
be perpendicular.
Planes can
be parallel.
intersecting
parallel
perpendicular
Similar Figures
Plane figures that have the
same shape are called similar
figures.
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Congruent Figures
Plane figures that have both the
same size and shape are called
congruent figures.
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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
Ray
A part of a line that has two
endpoints and thus has a definite
length is called a line segment.
A line segment extended
indefinitely in one direction is
called a ray.
Angle
Acute
Angle
Right
Angle
Obtuse
Angle
An angle is formed
when two rays share
the common point.
An angle whose
measure is less than 90
degree.
An angle whose
measure is 90 degree.
An angle whose
measure is greater than
90 and less than 180
degree.
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Straight
Angle
An angle whose measure
is 180 degrees.
Reflex
Angle
An angle whose measure
is greater than 180 and
less than 360 degree.
Plane
A plane is a flat surface
that has length and width
but no height.
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