Angles
In this article, we shall learn about angles. Angles play an essential part in Geometry. The plane figures that are developed at the vertex, where two endpoints are called angles. The angles are of different kinds. Now we shall discuss various kinds of angles with some example problems regarding angles.
Angles is a very important and vast concept of math. An angles is a geometric figure that has a common vertex and two arms as shown in the figures given below.
Fig.(i) Fig.(ii) Fig.(iii)
Let OA and OB be two half lines with common end point O. The half lines OA and OB are the sides of an angle and the point O is the vertex of the angle. An angle is an amount of rotation of a half-line (or ray) in a plane about its end point from an initial position to a terminal position.
Normally the angles are classified into the following types. They are
- Right angle
- Acute Angle
- Obtuse Angle
- Reflex Angle
- Complementary Angle
- Supplementary Angle
- Straight Angle
Now we shall have some discussion on these types of angles.
Right Angle:
If the angle measure of any object is exactly 900, then it is said to be right angle.
Acute Angle:
If the angle measure of any object is less than 900, then it is said to be acute angle.
Obtuse Angle:
If the angle measure of any object is greater than 900 and less than 1800, then it is said to be obtuse angle.
Straight Angle:
If the angle measure of any object is exactly 1800, then it is said to be straight angle.
Reflex Angle:
If the angle measure of any object is greater than 1800, then it is said to be reflex angle.
Complementary Angle:
If the sum of two adjacent angle measures exactly 900, then it is said to be complementary angle.
Supplementary Angle:
If the sum of two adjacent angle measures exactly 1800, then it is said to be supplementary angle.
Below are the examples on angles -
Example 1:
The angle measure of a shape is 1300. Determine what kind of angle it is?
Solution:
The given angle measure of the shape is 1300.
We know that the obtuse angle has the angle measures greater than 900 and less than 1800.
Therefore the given angle of the shape is an obtuse angle.
Example 2:
There are two adjacent angles. One of the angle measures 600. Determine the another angle, if the sum of two angles form a complementary angle.
Solution:
Given: One angle is 600.
We know that the sum of two angles measures exactly 900 to form a complementary angle.
Let us take the unknown angle as x0
60 + x = 90
Subtract 60 on both sides.
60 + x - 60 = 90 - 60
x = 300
Therefore the unknown angle is 300
Example 3:
Write complement of 420, 870, 200.
Angles 480, 30, 700 are the complements of the angles 420, 870, 200 respectively.
Example 4:
Write the supplement of 1080, 930, 1700.
Angles 720, 870, 100 are the supplements of the angles 1080, 930, 1700 respectively.
Example 5:
2x+112=360
2x=360-112=248
Construction of an angle equal to a given angle
and a point Q on a line PQ.
An angle equal to 
at Q.
Steps of Construction of an Angle:
(i) With B as centre and any suitable radius draw an arc to cut AB at D and BC at E.
(ii) With Q as centre and the same radius, draw an arc to cut PQ at S.
(iii) With S as centre and radius equal to DE draw an arc to cut this arc at T.
(iv) Join QT and produce it to R.
(Adjacent interior angles on the same side of a transversal are called allied angles).
If two parallel lines are cut by a transversal, then the allied angles are supplementary.
and 
Converse:
A transversal cuts two lines. If the allied angles are supplementary, then the lines are parallel.
If 
or 
, in the following figure then AB is parallel to CD.
| More topics in Angles | |
| Measuring Angles | Angle Pairs |
| Types of Angles | Bisector |
