1. What are angles and its types?
Angle is a measure of turn and is always measured in degrees(°). Degrees 30°, 60° 90°,120°, etc. show different angles.
On the basis of the size of the angle, you need to learn five main types of angles:
Acute angle: An angle measuring between 0° and 90° is called an acute angle.
Right angle: Angle measuring exactly 90° is called a right angle.
Obtuse angle: An angle measuring between 90° and 180° is called an obtuse angle.
Straight angle: An angle measuring exactly 180° is called a straight angle.
Reflex angle: An angle measuring between 180° and 360° is called a reflex angle.
2. Properties of Angles
You can be given some questions on finding the missing angles for 2D Shapes. In order to solve such questions, let us now learn about a few properties of angles along with a few solved examples.
2.1 Angles on a Straight line
Sum of all the angles on a straight line is always 180°.
Four angles a,b,c and d are lying on the straight line AB shown above.
Following the property, sum of all the angles on the straight line should be 180°.
a + b + c + d = 180°
2.2 Angles around a point
Sum of all the angles around a whole complete point is 360°.
Five angles a,b,c,d and e are around a point.
Following the property, sum of all the angles around point O should be 360°.
1 + b + c + d + e = 360°
2.3 Solved examples
Example 1 : Work out the value of angle marked x.
Solution: x, 32°, and 90° are the angles lying on straight line
Using the property of angles on a straight line,
Angles on a straight line add up to 180°
x + 32° + 90° = 180°
x + 122° = 180°
x = 180° - 122°
x = 58°
Example 2: Work out the value of angle marked x.
Solution: x, 40°,85° and 50° are the angles lying around a point.
Using the property of angles on around a point,
Angles around a point add up to 180°
x + 40° + 85° + 50° = 180°
x + 175° = 360°
x = 360° - 175°
x = 185°
3. Angles between hands of a Clock
In 11 Plus Exams, you may see questions on working out the angles between the minute hand and the hour hand of a clock at a given time. Let's learn about angles between the hands of a clock and understand it with a few examples.
A clock has 12 divisions as shown below,
Since the 12 divisions of the clock are equal, the angle between 12 and 1 is equal to the angle between 1 and 2 and so on.
What is the angle between each division?
The total angle or the angle for one complete turn is 360°, as angles around a point adds to 360°.
So angle between each division = 360° ÷ 12 = 30°
Now think of how much a minute hand moves in five minutes.
If the minute hand is at 12, after 5 minutes it will be at 1. So the minute hand would have turned by 30° because the angle between 12 and 1 is 30°.
If the hour hand is at 12, after 60 minutes it will be at 1. So hour hand would have turned by 30° because the angle between 12 and 1 is 30°.
3.1 Solved Examples
Example 1: What is the size of the angle marked a between the hands of the clock?
Solution: a is the angle between the hour hand and minute hand
Hour hand is at 5 and the minute hand is at 12
Angle between each division = 30°
From 12 to 5, there are 5 divisions.
Angle between 12 and 5
a = 5 × 30° = 150°
Example 2: On a clock face, what is the angle between the hands at 7.10?
Solution: At 7.10, the minute hand is at 2. Be careful about the hour hand.
The hour hand is not at 7, it has been 10 minutes since the hour hand
has moved past 7.
From 2 to 7, there are 5 divisions,
Angle between 2 and 7 = 5 × 30° =150°
In one hour or in 60 minutes, hour hands moves 30°.
In 10 minutes, the hour hand moves 10 ÷ 60 × 30° = 5°
From 2 to 7, the angle between the hour hand and minute hand at
7.10 = 150° + 5° = 155°
