Angles are those which are formed by two rays with common vertex. If one of the ray is fixed and the other is rotated about the fixed vertex, then the angle between the two rays will measure from 0o to 360o. The angles can be classified according to their measure. Measuring angles is the part of practical geometry. We measure the angles to classify into the type of the angles. The angles can be measured to find the angles subtended according to the given condition.
For example, angle subtended by an object at the retina, angle of elevation of the top of a building, angle of depression of a ship from the top of a hill etc. Measurement of these angles will help us to find the top of a building, distance between the buildings, distance of the ship from the hill or shore etc. In this section let us see how we measure angles.

## How to Measure Angles

What are angles? Angles are those which are formed with two rays called as arms with a common vertex.
When one of the arm is fixed and when we rotate the other arm about the fixed vertex, the different types of angles are formed.The angles are named either by writing the vertex or by using the end points on the arms as shown below.

The following diagrams show the $\angle A$ or $\angle PAQ$

### Classification of Angles

Angles can be classified according to their measure.
We can see the definition of each type of angle under the figure given below which describes the types of angles.

Let us see the definitions of the types of angles.

1. Zero Angle: When the measure of the angle is 0 o, that is when the two arms coincide, the measure of the angle is 0o ( read as zero degree ).

2. Acute angle: When the angle measure more than 0 o and less than 90 o, then the angle is called an acute angle.Example: 20o , 1o , 899 .
3. Right angle: When the angle measure exactly 90o, then the angle is called a right angle or we call that the two arms are perpendicular to one another.

4. Obtuse angle: When the measure of the angle is more than 90o and less than 180o , then the angle is called an obtuse angle.
Example: 91o , 100o , 179o
5. Straight angle: When the angle measure exactly 180o , then the angle is called a straight angle.

6. Reflex angle: If the measure of angle is more than 1800 , and less than 360o , then the angle is called reflex angle.
Example: 181o , 200o, 300o , 359o.
7. Complete angle: An angle whose measure is 360o is called a complete angle. In a complete angle the moving arm would have completed one rotation about the fixed point and coincide with the fixed arm, like the zero angle.

## Tool to Measure Angles

The angles can be measured using a tool called protractor. As the angle along a straight line is 180o, the protractor shows 180o from either side of it.

The following diagram shows a protractor.

The definite angles like 45o , 30o , 60o and 90o can be measured using the geometrical tool called "Set Square".

The diagram above show the set squares with definite angles. The same can be placed on a given angle to identify if the angle measures exactly any of these of greater than or less than these angles.

## Measuring Angles with a Protractor

When we measure angles using the protractor, we should remember the following steps.
Step 1: Place the center of the protractor at the vertex of the angle.
Step 2: Make sure that one of the horizontal line is along one of the arm of the angle.
Step 3: Slowly move your eyes from the zero of the protractor which is along the line in step 2.
Step 4: Look for the angle measure which coincide with the other arm of the angle drawn. This will give the exact measure of the angle.

## Measuring Angles Without a Protractor

Set Squares: First we need to place the edge containing any of the angles of a set square on the angle so that one of the edge of the set square is along one of the arm of the angle. Then look for the other arm if it coincides with the other edge of the set square or not.

1. The set squares shown above can be used to identify if the angle measures, 45o , 30o , 60o or 90o .
2. Set squares will also help us to identify if the angle measures greater than or less than 45o , 30o , 60o or 90o .

Trigonometry:
1. We can use the trigonometrical functions and their ratios to find the exact measure of the angle by drawing a perpendicular from one arm onto another.

We should also find the length of the base, perpendicular and the hypotenuse (which is the length of the other arm from which the perpendicular is drawn).

From the above figure, we can measure the angle, $\angle AOC$ using the trigonometric functions. The following steps will help us to evaluate the measure of the $\angle AOC$.
Step ( i ): Draw a perpendicular AM on the arm OB
Step ( ii ). Measure the length of AM and OA.
Step ( iii ). Use the trigonometric ratio sin $\theta$ = $\frac{Opposite}{Hypotenuse}$ = $\frac{AM}{OA}$
Step ( iv ):. Use the trigonometric table containing Natural Sines, to look for the value shown above in step 3, and find the corresponding angle which is $\theta$.

2. Law of cosines:
Measuring the angles of a triangle using Law of cosines, when the length of the sides of the triangle are given.
Let us assume that the length of the sides of the triangle ABC are, a, b, c.

We can use the following formulae, to evaluate the angles of the $\Delta ABC$
Cos A = $\frac{b^{2}+c^{2}-a^{2}}{2bc}$

Cos B = $\frac{a^{2}+c^{2}-b^{2}}{2ac}$

Cos C = $\frac{b^{2}+a^{2}-c^{2}}{2ab}$

## Measuring Angles Practice

1. Measure the following angle $\angle PQR$ using protractor and classify it into its type.

2. Use Set square of classify if the following angles are acute or obtuse.

3. Use trigonometric function to evaluate the angle $\angle KLM$.