Angles, lines, rays, and segments

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Angles, lines, rays, and segments
MATHEMATICS
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Contents

• Recap the terms

• Test Yourself - 1

• Angles in daily life

• Congruent angles

• Pairs of angles Types

• What is an angle?

• Test Yourself - 2

• Naming an angle

• Pairs of angles formed by a transversal

• Interior and exterior of an angle

• Test Yourself - 3

• Measurement of angle

• Types of angle Right angle
• Obtuse angle
• Acute angle
• Straight angle

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Recap Geometrical Terms




An exact location on a plane is called a point.
Point
A straight path on a plane, extending in both
directions with no endpoints, is called a line.
Line
A part of a line that has two endpoints and thus
has a definite length is called a line segment.
Line segment
A line segment extended indefinitely in one
direction is called a ray.
Ray
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Angles In Daily Life
If we look around us, we will see angles
everywhere.
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What Is An Angle ?
When two non-collinear rays join with a common
endpoint (origin) an angle is formed.
Ray BA
B
Common endpoint
Ray BC
Common endpoint is called the vertex of the
angle. B is the vertex of ÐABC.
Ray BA and BC are two non-collinear rays
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Fact We can also think of an angle formed by
rotating one ray away from its
initial position.
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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.
For example ÐABC or ÐCBA
We may also name this angle only by the single
letter of the vertex, for example ÐB.
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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)

• Points within the angle (Its interior portion. )

• Points outside the angle (Its exterior portion.
  )

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Measurement Of An Angle
Protractor is used to measure and draw angles.
Angles are accurately measured in degrees.
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Types Of Angles
There are four main types of angles.
Right angle
Acute angle
Obtuse angle
Straight angle
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Right angle An angle whose measure is 90
degrees.
Right Angle
Acute Angle
Straight Angle
Obtuse Angle
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Examples Of Right Angle
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Obtuse angle An angle whose measure is greater
than 90 degrees.
Right Angle
Acute Angle
Straight Angle
Obtuse Angle
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Examples Of Obtuse Angle
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Acute angle An angle whose measure is less than
90 degrees.
Right Angle
Acute Angle
Straight Angle
Obtuse Angle
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Examples Of Acute Angle
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Straight angle An angle whose measure is 180
degrees.
Right Angle
Acute Angle
Straight Angle
Obtuse Angle
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Examples Of Straight Angle
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Which of the angles below is a right angle, less
than a right angle and greater than a right angle?
1.
2.
Test Yourself 1
Greater than a right angle
3.
Right angle
Less than a right angle
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Congruent Angles
Two angles that have the same measure are called
congruent angles.
Congruent angles have the same size and shape.
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Pairs Of Angles Types

• Adjacent angles

• Vertically opposite angles

• Complimentary angles

• Supplementary angles

• Linear pairs of angles

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Adjacent Angles
Two angles that have a common vertex and a common
ray are called adjacent angles.
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 pairs of angles
formed by two lines intersecting at a point.
ÐAPC ÐBPD
ÐAPB ÐCPD
P
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.
ÐABC and ÐDEF are complimentary because
ÐABC ÐDEF
600 300 900
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Contd.
If the sum of two angles is more than 900 or
less than 900, then they not complimentary angles.
ÐDEF and ÐPQR are not complimentary because
ÐDEF ÐPQR
700 300 1000
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Supplementary Angles
If the sum of two angles is 1800 then they are
called supplementary angles.
ÐPQR and ÐABC are supplementary, because
ÐPQR ÐABC
1000 800 1800
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Contd.
If the sum of two angles is more than 1800 or
less than 1800, then they are not supplementary
angles.
ÐDEF and ÐPQR are not supplementary because
ÐABC ÐDEF
1100 800 1900
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Linear Pair Of Angles
Two adjacent supplementary angles are called
linear pair of angles.
P
ÐAPC ÐAPD
600 1200 1800
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Name the adjacent angles and linear pair of
angles in the given figure
Adjacent angles
ÐABD and ÐDBC
ÐABE and ÐDBA
Test Yourself 2
Linear pair of angles
ÐEBA, ÐABC
ÐEBD, ÐDBC
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Name the vertically opposite angles and adjacent
angles in the given figure
Vertically opposite angles ÐAPC and ÐBPD
Adjacent angles ÐAPC and ÐCPD
ÐAPB and ÐCPD
ÐAPB and ÐBPD
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Pairs Of Angles Formed by a Transversal
A line that intersects two or more lines at
different points is called a transversal.
Line L (transversal)
Line M
P
Line N
Q
Four angles are formed at point P and another
four at point Q by the transversal L.
Line M and line N are parallel lines.
Line L intersects line M and line N at point P
and Q.
Eight angles are formed in all by the 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 parallel lines are cut by a transversal,
pairs of corresponding angles are formed.
ÐGPB ÐPQE ÐGPA ÐPQD ÐBPQ ÐEQF ÐAPQ ÐDQF
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 transversal and at different intersecting
points.
ÐBPQ ÐDQP
ÐAPQ ÐEQP
Two pairs of alternate angles are formed.
Pairs of alternate angles are congruent.
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Interior Angles
The angles that lie in the area between the two
parallel lines that are cut by a transversal, are
called interior angles.
ÐBPQ ÐEQP 1800
600
1200
1200
600
ÐAPQ ÐDQP 1800
The measures of interior angles in each pair add
up to 1800.
A pair of interior angles lie on the same side of
the transversal.
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Name the pairs of the following angles formed by
a transversal.
Test Yourself 3
Corresponding angles
Alternate angles
Interior angles
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The End