An angle is formed by two rays with a common end point. The rays forming the angle are called the sides of the angle and the common end point is known as the vertex of the angle.

What is Angle

The sides of the angle are AB and AC.
The vertex is A.
How to name the angle?

We can name the angle
  • by its vertex ∠A.
  • Using a point on each side and the vertex,as ∠BAC or ∠CAB.
  • Using a number ∠1.
 


Degree is the unit used to represent an angle measure and it denoted by the symbol º. A full angle formed around a point measures 360º. The angle at point are classified based on their measures as follows:

Type 
 Degree measure
 1. Acute angle  0º < x < 90º
 2. Right angle   x = 90º
 3. Obtuse angle  90º < x < 180º
 4. Straight angle  x =180º
 5. Reflex angle  180º < x < 360º

Let us see few examples on each of the above mentioned type.

The measure of an acute angle is greater than 0º and less than 90º.
Using math, if x is the measure of an acute angle then 0º < x < 90º.

If the angle measure x + y = 90º, then y is said to be the complementary measure of the angle whose measure is x.

Thus the complementary angle of an acute angle is also an acute angle.

Solved Example

Question:
Acute Angle

In the above diagram, the two angles 1 and 2 are complementary angles. Find the measure of each angle.
Solution:
 
measure of 1 + measure of 2 = 90                                             Definition of complementary angles
x + 3 + 2x - 3 = 90
                 3x = 90                                                                      Combining the like terms
                   x = 30º.
Measure of angle 1 = 2x - 3 = 2(30) - 3 = 60 - 3 = 57º.
Measure of angle 2 = x + 3 = 30 + 3 = 33º.

As the sum of three angles in a triangle is 180º, a triangle has at least two acute angles.
A triangle with all the three angles as acute is called an acute angled triangle.
 

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A right angle measures exactly 90º. If A is a right angle then m ∠A = 90º.

A right angle is formed when the two sided forming the angle are perpendicular to each other. Indeed when two perpendicular lines intersect, four right angles are formed at the point of intersection.
A right angle is denoted by square type symbols like '⌈' or '⌉'.
Right Angle

Thus, the angle formed at the foot of a perpendicular is a right angle.

A right angle can have at most one right angle and such triangles are called the right triangles. Right triangles are used in solving many real life situations using trigonometry.

All the four angles in a rectangle or a square are right angles.While the diagonals of a rhombus bisect each other at right angles, the longer diagonal of a kite
bisects the shorter diagonal at right angles.

Kite

Right angles are the most observed angles in the objects around us.
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The measure of an obtuse angle is greater than 90º and less than 180º. If ∠A is an obtuse angle, this can be mathematically expresses as 90º < m ∠A < 180º.

Obtuse Angle

In the above diagram the angle BAC is greater than the right angle marked by a square sign.
m ∠BAC > 90º.

Indeed the obtuse angle is formed by adding an acute measure to the right angle.
Angle 2 which is obtuse makes a straight angle with the acute angle 1.
Thus 1 and 2 are supplementary angles. In general the supplement of an obtuse angle is acute.
The complement for an acute angle does not exist, as its measure is greater than 90º.

When two non perpendicular lines intersect one pair of equal vertical angles are obtuse.

Obtuse Acute

In general whenever the two sides deviate wide, an obtuse angle is formed. When the laptop is opened for working, the screen makes an obtuse angle with the base.
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Straight angle is an angle that is the union of two opposite rays. A straight angle measures exactly 180º.
Unlike the angles acute, right and obtuse angles which are formed at the intersection of non-collinear rays, the straight angle is formed by collinear but opposite rays.

Straight Angle

In the above diagram, the straight angle ABC is formed by the two opposite rays BA and BC.
The line perpendicular to AC at B, BD makes two right angles at B. Arithmetically also the measure of a straight angle is made up of two right angles.

180º = 2 x 90º.
Straight angles are formed when folding objects are spread open, like the angle between the open pages of book kept fully open.
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Reflex angles are made on the exterior of an acute, a right or an obtuse angle. The measure of a reflex angle is greater than 180º, but less than 360º.

Reflex Angle
In the case of a straight angle, the angles formed on either side of the vertex are straight angles.

An angle and the reflex angle at a point make up the full angle which measures 360º. Hence the measure of the reflex angle of an angle is got by subtracting the measure of the angle from 360º.
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