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.l8

Two rays \overrightarrow{OA} and \overrightarrow{OB}  and  having a common point will form an angle AOB which is written as \angle AOB . OA  andOB  are called the arms of the angle and O is called the vertex of the angle \angle AOB .

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 {\ }^{\circ} .

A complete rotation around a point makes an angle of 360^{\circ} .l9

One degree 1^{\circ}=60 minutes (also written as 60^{'} ).

One Minute1^{'}=60  seconds (also written as 60^{''} ).

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 0^{\circ}  but less than 90^{\circ}  is called an acute angle. l10
Right Angle An angle whose measure is equal to 90^{\circ}  is called a right angle. l11
Obtuse Angle An angle whose measure is more than 90^{\circ}  but less than 180^{\circ}  is called an obtuse angle. l12
Straight Angle An angle whose measure is equal to 180^{\circ}  is called a straight angle. l13
Reflex Angle An angle whose measure is more than 180^{\circ}  but less than 360^{\circ}  is called a reflex angle. l14
Complete Angle An angle whose measure is equal to 360^{\circ}  is called a complete angle. l15

Equal Angles: Two angles are said to be equal if they have the same measure.l16

Bisector of an Angle: Any ray is called a bisector of an angle if \angle AOC=\angle COB .

Complimentary Angles: If the sum of two angles is 90^{\circ} , 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 180^{\circ} , 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 wayl17 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 O  is the common vertex, and OC  is the common arm. Hence we can say that \angle AOC \ \&\ \angle COB  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 \angle AOC+\angle COB=180^{\circ}=\angle AOC

Another way of looking at this is that is the sum of two adjacent angles is 180^{\circ} , 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 360^{\circ} .

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

As you see, lines AB\ and\ CD intersect at point O . It forms two pairs of vertically opposite angles, which are:l19

\angle AOD \ \& \ \angle COB  are vertically opposite

\angle AOC \ \&\  \angle DOB  are vertically opposite

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

Given: Line  AB \ and\  CD   intersect at point  O 

To Prove: i) \angle BOC=\angle AOD and ii) \angle AOC=\angle DOB

Proof: Since ray OC stands on a straight line

\angle AOC+\angle BOC = 180^{\circ}           [Linear Pair Axiom]

Similarly, since ray  OA  stands on line  CD 

\angle AOC+\angle AOD=180^{\circ}          [Linear Pair Axiom]

Therefore \angle AOC+\angle BOC=\angle AOC+\angle AOD

Or \angle BOC=\angle AOD . Hence provedl20

Similarly, you can prove \angle AOC=\angle DOB

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