Download Class 8: Chapter 28 – Lines and Angles (Lecture

Class 8: Chapter 28 – Lines and Angles
(Lecture Notes) – Part 2
Angles: an angle is the figure formed by two rays, called
the sides of the angle, sharing a common endpoint, called
the vertex of the angle.
⃗⃗⃗⃗⃗ and 𝑂𝐵
⃗⃗⃗⃗⃗ having a common point will form an angle
Two rays 𝑂𝐴
AOB which is written as ∠AOB. OA and OB are called the arms of the angle and O is called
the vertex of the angle ∠AOB.
Measure of an Angle: It is the amount of rotation through
which one arm of the angle has to be rotated, about the vertex,
to bring it to the position of the other arm.
Angle is measured in degrees, denoted by °.
A complete rotation around a point makes an angle of 360°.
One degree (1°) = 60 minutes (also written as 60´).
One Minute (1´) = 60 seconds (also written as 60´´).
To draw angles, the commonly used equipment is called protector.
Kinds of Angles
Name of the
Angle
Description
Acute Angle
An angle whose measure is
more than 0° but less than 90°
is called an acute angle.
Diagram
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Right Angle
An angle whose measure is
equal to 90° is called a right
angle.
Obtuse
Angle
An angle whose measure is
more than 90° but less than
180° is called an obtuse angle.
Straight
Angle
An angle whose measure is
equal to 180° is called a
straight angle.
Reflex
Angle
An angle whose measure is
more than 180° but less than
360° is called a reflex angle.
Complete
Angle
An angle whose measure is
equal to 360° is called a
complete angle.
Equal Angles: Two angles are said to be equal if they have the
same measure.
Bisector of an Angle: Any ray is called a bisector of an angle if
∠AOC = ∠COB
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Complimentary Angles: If the sum of two angles is 90°, then the angles are called
complimentary angles. We can also say that Complementary angles are angle pairs whose
measures sum to one right angle.
Supplementary Angles: If the sum of two angles is 180°, then the angles are called
supplementary angles. If the two supplementary angles are adjacent
their non-shared sides form a straight line.
Adjacent Angles: If two angles share one common arm and a
common vertex in such a way that the other angle arms are on either
side of the common arm then they are called adjacent angles. In this
example we see that O is the common vertex, and OC is the common arm.
Hence we can say that ∠AOC & ∠COB are adjacent angles.
Linear Pair of Angles: If the adjacent angles are such that the, the non-common arms
form a straight angle, then the angles are called
linear pair of angles. In this case ∠AOB + ∠COB =
180° = ∠AOC
Another way of looking at this is that is the sum of
two adjacent angles is 180°, then they will form a
linear pair of angles.
One more important result that you should know is that the sum of angles around a point
(or dot) is 360°.
Vertically Opposite Angles: When two straight lines intersect at one point, they will
form vertically opposite angles which are equal.
As you see, lines AB and CD intersect at point O. It forms two pairs of vertically opposite
angles, which are:
∠AOD & ∠COB are vertically opposite
∠AOC & ∠DOB are vertically opposite
We can also prove that these angles are equal to each
other.
Given: Line AB and CD intersect at point O
To Prove: i) ∠BOC = ∠AOD and ii) ∠AOC = ∠DOB
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Proof: Since ray OC stands on a straight line
∠AOC + ∠BOC = 180°
[Linear Pair Axiom]
Similarly, since ray OA stands on line CD
∠AOC + ∠AOD = 180°
[Linear Pair Axiom]
Therefore ∠AOC + ∠BOC = ∠AOC + ∠AOD
Or ∠BOC = ∠AOD. Hence proved
Similarly, you can prove ∠AOC = ∠DOB
Perpendicular Lines: A line is said to be perpendicular to
another line if the two lines intersect at a right angle. If AB and
CD are two perpendicular lines, then they are denoted as AB ⊥
CD.
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