Angles

Ray: A part of line, with one end point and the other extending in one direction is called a ray.
Ray

A ray is named according to the direction in which it extends. In the above figure the end point is B and the line is extending in the direction of AB and hence the ray is named as ray AB and denoted as $\overrightarrow{AB}$ .

Let us consider another example,
Ray Example

In the above figure, the ray has B as its end point and the line extending in the direction of A. It is denoted by $\overrightarrow{BA}$

Angle is formed when two rays originate from the same endpoint.
The two rays are called the arms of the angle and the endpoint at which the two rays meet is called a vertex. Some of the angles are given in the figures below:
Vertex

Types of Angles

There are different types of angles

1.    Acute angle: acute angle is the angle which is less than 90° and greater than 0°. The angle lies between 0° and 90°.
Acute Angle
The above figure shows an acute angle with angle measure $\theta$ 

Acute angle : 0°< $\theta$< 90°

2.     Right angle: It is an angle with measure exactly 90°. The two arms of the angle are perpendicular to each other.
Right Angle
  In this figure the angle $\theta$=90°, which is the right angle

3.    Obtuse Angle: It is an angle measuring between 90° and 180°.
Obtuse Angle
 In the given figure the measure of angle is θ,  the obtuse angle is 90°< $\theta$ <180°

4.    Straight Angle: As the name suggests the measure of the angle is that of a straight line, which is 180°
 Straight Angle
The measure of the angle θ in the above figure is  $\theta$ =180°, the straight angle.

5.    Reflex angle: The measure of the angle which is greater than 180° and less than 360° is called the reflex angle
 Reflex Angle
The measure of the angle $\theta$¸ in the above figure is 180° < $\theta$ < 360°, the reflex angle

6.    Complementary angles: If the sum of the measures of two angles is equal to 90 degrees, the two angles together are called complementary angles. Each angle complement the other in complementary angles
For example: $\angle$A = 60° and $\angle$B = 30° 

Sum of the angles, $\angle$A + $\angle$B = $\angle$60° + $\angle$30° 
                                                                              = 90°

Since the sum of the angles is equal to 90 degrees, $\angle$A and $\angle$B are called the complementary angles. $\angle$A complements the $\angle$B and vice versa.
 Supplementary Angle
7.    Supplementary Angles: If the sum of the measures of two angles is equal to 120 degrees, the two angles together are called complementary angles. Each angle supplement the other in supplementary angles.

For example: $\angle$P = 110° and $\angle$Q = 70° 
Sum of the angles, $\angle$P + $\angle$Q = 110° + 70°
                                 = 180°
Since the sum of the angles is equal to 180 degrees, $\angle$P and $\angle$Q are called the supplementary angles. $\angle$P supplements the $\angle$Q  and vice versa.
 Vertical Angles

8.    Vertical Angles: The angles formed when two lines intersect each other are called vertical angles. They have the same measure and hence they are equal.
Vertically Opposite Angles
 
In the figure given above the lines PQ and RS intersect at the point T. $\angle$PTR and $\angle$STQ are one set of vertical angles and are equal in measure. $\angle$PTS and $\angle$RTQ are another set of vertical angles and are equal in measure. Vertical angles are also called as vertically opposite angles.

Measuring Angles

Measuring angles is made easy using an instrument called 'Protractor'. 
 Measuring Angles
An angle is measured usually using an instrument, a protractor. It looks like the figure shown above. Each unit of the measure on the protractor is equal to 1°, it totally consists of 180° (180 divisions) shown from both the sides starting with 0°.
 Protractor Measure
Angle can be measured from either sides of the protractor according to the orientation of the protractor.

Let us now learn how to measure an angle using a protractor:
Let us find the measure of the angle PQR as given in the figure below,
 Protractor Measure Example
 To measure the given angle of PQR, first we need to place the protractor in such a way that the centre of the protractor coincides exactly with the vertex (Q) and the base line of the protractor is along the arm QP, of the angle PQR. We shall use the inner measure of the protractor (given angle is less than 90 degrees ,  an acute angle) to measure the angle PQR, as the arm PQ passes through the zero of the inner scale. Now, we need to follow the inner scale around the protractor to find the other arm, QR.
The arm QR passes through 50° in the inner scale. So, the measure of the angle PQR is 50°, we write the angle as follows:
$\angle$PQR = 50° Example: Let us measure the obtuse angle ABC as given in the figure below using a protractor
 Measure
To measure the given angle ABC, place the protractor in such a way that the base of the protractor passes through the arm  BA and the centre of the protractor coincides exactly with the vertex B of the angle ABC. We shall use the outer scale to measure of the angle ABC, as the arm AB passes through the zero of the outer scale. We now need to follow the outer scale around the protractor, we find that the other arm BC, passes through the outer scale at 130°. So, finally we get the measure of the angle ABC as 130°.  We write the angle as
            $\angle$ABC = 130°

Parts of Angles

Angle
    
An angle consists of three main parts:
Rays: the two arms of the angles
Vertex: the intersecting point (endpoint) of the two     arms (rays) of the angle
Angle value: The rotation required to bring one of the arms to coincide with the other arm will give the angle value.

Positive and Negative Angles

Positive angle: An angle when rotated in the anti-clockwise direction the angle generated is a positive angle.
  Positive Angle 

Negative angle: An angle when rotated in the clockwise direction, the angle generated is a negative angle.
Negative Angle

Identifying Angles

Let us identify the following angles given:
1. Obtuse Angle Example
Solution: In the above figure, Angle A is greater than 90 degrees and less than 180 degrees. Hence
∟A is an Obtuse angle.


2. Identify the pair of angle P & Q and R & S in the given figure
 Angles Example
Solution: In the given figure, the angle P and Q are the vertical angles  and the angles R and S are the supplementary angles, the sum of these angles is 180 degrees (straight angle).


3. Identify the given pair of angles in the figure below
Pair of Angle 
Solution: In the given figure the angles A and B are each 45°. Sum of these angles is equal to 90° and hence they complement each other
Angle A and Angle B are complementary angles.


4. Identify the type of angle in the figure given
Reflex Angle Example
Solution: The angle shown in the figure is a reflex angle.


5. Identify the angle shown in the figure
 Acute Angle Example
Solution: The angle shown in the figure is an Acute angle  as it is more than zero degrees and less than 90 degrees.

Labeling Angles

Angles can be labeled either by using lower-case letters like p or q. They can be labeled even by using the greek letters like θ (theta), α (alpha) or Î2 (beta)
They can at times labeled using three letters of a shape where the angle is defined, with the letter in the middle showing the actual angle, which is the vertex.
For example, In the figure given below, $\angle$QPS = a is labeled by a lower-case letter 'a' and the $\angle$PQR = $\theta$ is labeled by a greek letter $\theta$.
 Actual Angle
The Q can also be labeled by using three letters $\angle$PQR, the actual angle being Q, the vertex
So, in the above figure, $\angle$PQR is  '$\theta$' and $\angle$SPQ is 'a'  (here Q and P are two vertices of the figure given).