ANGLES
Definition
Types of angles
Parts of angles
Positive and Negative angles
Types of angles according to position
Properties
Classification of angles
Identification of angles.
CLASSIFICATION OF ANGLES:
• Example 1: reflex angles
• The lines A and B meet at the point O. Calculate the missing angle a.
• Step 1: Add all known angles.= 1280
• Step 2: Subtract the angle sum from 360°.
3600−1280 = 2320
•
a0 = 2320
(a) ANGLES AT A POINT
(b) ADJACENT ANGLES
(c) SUPPLEMENTARY ANGLES
(d) COMPLEMENTARY ANGLES
(e) VERTICALLY OPPOSITE ANGLES
• Types of angles according to position
• Some angles are given special names because of their
positions.
(a)Angle at a point: This when two or more angles have
the same vertex they are known as angles at a point.
Angles a, b, c, d and e are all angles at a point.
∠a + ∠b + ∠c + ∠d + ∠e = 3600
i.e., 3600 is one complete turn.
The angle at a point always add up to 3600.
How to find missing angles around a point
In order to find missing angles around a
point:
1.Add all known angles.
2.Subtract the angle sum from 360°.
3.Form and solve the equation.
Examples
SINGLE ANGLE AROUND A POINT : REFLEX ANGLE
• Example 1: reflex angles
• The lines A and B meet at the point O. Calculate the missing angle a.
Step 1: Add all known angles.
• 1280
Step 2: Subtract the angle sum from 360°.
• 3600 − 1280 = 2320
a0 =2320
•
Example 2: multiple angles around a point
Calculate the size of angle x.
Solution:
Step 1:
Add all known angles = 1510 + 640 = 2150
Step 2:
Subtract the angle sum from 3600
3600 - 2150 = 1450
x0 = 1450
•
Class work
• Forming and Solving Equations:
•
HOMEWORK
•(b) Adjacent Angles:
•This is when two angles at appoint
have one arm in common and lie on
opposite sides of that arm, they are
known as adjacent angles. Angles b
and c are adjacent angles.
•Angles b and c are adjacent angles.
•Class work: Identification of Adjacent Angles:
• Calculation Involving Adjacent Angles:
• Example 1:
• Note : when two Adjacent angles add up to 1800, they are known as
Supplementary Angles. But when two adjacent angles add up to 900
they are known as Complementary Angles.
• The example below is supplementary because its add up to 1800
• Find ∠ROP
• Solution:
• Add all the known angles
• = 800 + 360 = 1160
• Therefore ∠ROP = 1160
• Find ∠DAB ?
• Find ∠XWZ ?
•(c) Supplementary angles:
•This is when two adjacent angles add up to
0
180 , they are known as supplementary
angles.
•Angles c and are supplementary because LM
0
is a straight line. ∠c + ∠d = 180 .
•Angles c and d are also known as angles on
a straight line.
•
•
IDENTIFICATION OF
SUPPLEMENTARY ANGLES
Calculation Involving Supplementary Angles:
Supplementary Angles Examples:
Example 1:
• Find the value of angle Y in the following figure.
• Solution:
• In the given figure, Y and 77° are supplementary angles as
they lie at a point on a straight line. Hence, their sum is 180°.
• Y + 77° = 180°
• Y = 180° - 77° = 103°
• Therefore, Y = 103°
• Example 2:
• Find the values of Angle A and Angle B, if they are supplementary
angles such that ∠A = (2x + 10)° and Angle B = (6x − 46)°
• Solution:
• Since Angle A and Angle B are supplementary angles, their sum is 180°
• ∠A + ∠B = 180°
• (2x + 10) + (6x - 46) = 180°
• 8x - 36 =180
• 8x = 216
• x = 27
• Therefore, Angle A = 2(27) + 10 = 64° and Angle B = 6(27) - 46 =116°
• Example 3: Diagram for Illustration
Example 3:
Find the value of x if the following two angles are supplementary.
Solution:
Since the given angles are supplementary, their sum is 180°.
𝑥 𝑥
+ =180°
2
3
5𝑥
5𝑥
180
=180°=
=
(cross multiply)
6
6
1
5x × 1 = 6 x 1800
5x = 6 x 180
X
6 ×180
=
, x = 2160
5
• This means one angle is 216/2 = 108° and the other angle is 216/3 = 72°.
• They can be verified as supplementary angles because their sum is 180°, that is,
108° + 72° = 180°
•Practice Questions on Supplementary Angles
•Q1: Select the pair of supplementary angles
from the following.
0
0
•115 , 65
0
0
•120 , 115
0
0
•15 , 65
•1700 , 1000
Q2: Select the pair of supplementary
angles from the following.
0
0
14 , 175
0
0
18 , 162
0
0
11 ., 179
0
0
21 , 163
• Q. 3
• Choose 2 correct options from the following.
• If ∠A and ∠B are supplementary angles, then which of these can be true?
• Both ∠A and ∠B are acute angles.
• Both ∠A and ∠B are obtuse angles.
• Both ∠A and ∠B are right angles.
• ∠A is an acute angle and ∠B is an obtuse angle.
• Find the missing angle measurement in each set of supplementary angles
•(d) Complementary Angles:
•This when two adjacent angles add up to
0
90 , they are known as complementary
angles.
•Angles a and b are complementary angles.
•Angles a and b are complementary because
0
∠COB is a right angle. ∠a + ∠b = 90 .
• Above we have two non-adjacent angles, ∠K∠K and ∠N∠N.
But since the m∠K= 480 m∠K=480 and m∠N=420
m∠N=420 add up to 900 , they are also called complementary
angles.
• Example 2:
• Angle CYA = 180 + X = 900
• Note that complementary angles add up to 900
• = 180 + x = 900
• X = 900 – 180 = 720
• Check: 180 + 720 = 900
•
CLASS WORK
• Find the missing angle measurement in each set of complementary angles
• (e) Vertically Opposite Angles:
• This is when two or more straight lines cross each other
forming four or more angles at a point, then the pairs of
angles opposite to each other are known as vertically
opposite angles.
• angle a = angle b
• angle p = angle q
• Vertically opposite angles are always equal to each
other.