Angles: an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle.

Two rays and and having a common point will form an angle AOB which is written as . and are called the arms of the angle and O is called the vertex of the angle .

Measure of an Angle: It is the amount of rotation through which one arm of the angle has to be rotated, about the vertex, to bring it to the position of the other arm.

Angle is measured in degrees, denoted by .

A complete rotation around a point makes an angle of .

One degree minutes (also written as ).

One Minute seconds (also written as ).

To draw angles, the commonly used equipment is called protector.

Kinds of Angles

Name of the Angle | Description | Diagram |

Acute Angle | An angle whose measure is more than but less than is called an acute angle. | |

Right Angle | An angle whose measure is equal to is called a right angle. | |

Obtuse Angle | An angle whose measure is more than but less than is called an obtuse angle. | |

Straight Angle | An angle whose measure is equal to is called a straight angle. | |

Reflex Angle | An angle whose measure is more than but less than is called a reflex angle. | |

Complete Angle | An angle whose measure is equal to is called a complete angle. |

Equal Angles: Two angles are said to be equal if they have the same measure.

Bisector of an Angle: Any ray is called a bisector of an angle if .

Complimentary Angles: If the sum of two angles is , then the angles are called complimentary angles. We can also say that Complementary angles are angle pairs whose measures sum to one right angle.

Supplementary Angles: If the sum of two angles is , then the angles are called supplementary angles. If the two supplementary angles are adjacent their non-shared sides form a straight line.

Adjacent Angles: If two angles share one common arm and a common vertex in such a way that the other angle arms are on either side of the common arm then they are called adjacent angles. In this example we see that is the common vertex, and is the common arm. Hence we can say that are adjacent angles.

Linear Pair of Angles: If the adjacent angles are such that the, the non-common arms form a straight angle, then the angles are called linear pair of angles. In this case

Another way of looking at this is that is the sum of two adjacent angles is , then they will form a linear pair of angles.

One more important result that you should know is that the sum of angles around a point (or dot) is .

Vertically Opposite Angles: When two straight lines intersect at one point, they will form vertically opposite angles which are equal.

As you see, lines intersect at point . It forms two pairs of vertically opposite angles, which are:

are vertically opposite

are vertically opposite

We can also prove that these angles are equal to each other.

Given: Line intersect at point

To Prove: i) and ii)

Proof: Since ray stands on a straight line

[Linear Pair Axiom]

Similarly, since ray stands on line

[Linear Pair Axiom]

Therefore

Or . Hence proved

Similarly, you can prove

Perpendicular Lines: A line is said to be perpendicular to another line if the two lines intersect at a right angle. If are two perpendicular lines, then they are denoted as .