In order to measure angles, we must first select some fixed unit. The natural unit would be a right angle. This, however, is an inconvenient unit of measurement on account of its size. Thus, for convenience two systems of measurement have been established, in each of which the unit is a certain fraction of a right angle. The two systems of measurement are:

Sexagesimal Measure (English System): In this system of measurement, a right angle is divided into 90 equal parts called Degrees; a degree is divided into 60 equal parts called Minutes, and a minute into 60 equal parts called Seconds.

The symbols $1^o$, 1', 1" are used to denote a degree, a minute, and a second respectively. Thus,

60" = 1'

60' = $1^o$

$90^o$ = 1 Right Angle

**Examples:**

$91^o$ 20' 58", $18^o$ 21', and $100^o$ 0' 28" Centesimal Measure (French System): In this system of measurement, a right angle is divided into 100 equal parts called Grades, a grade is divided into 100 equal parts called Minutes, a minute into 100 equal parts called Seconds.

The symbols $1^g$, 1', 1" are used to denote a degree, a minute, and a second respectively. Thus,

100" = 1'

100' = $1^g$

$100^g$ = 1 Right Angle

**Examples:**

$100^g$ 25' 80", $105^g$ 0' 92", and $108^g$ 22'Even though the same names are used, a centesimal minute and second are not the same as a sexagesimal minute and second. Thus, a right angle contains 90×60=5400 sexagesimal minutes, whereas it contains 100 × 100 = 10000 centesimal minutes.

Circular Measure (Radian Measure): This is the third system of measurement which is using in all the higher branches of Mathematics. The unit used in this system of measurement is a radian, denoted by $1^c$, and is defined as the angle subtended at the center of any circle by an arc equal in length to the radius of the circle.

Consider a circle with center O and radius r units. Let $\widehat{AB}$ be an arc with length r units. Then, the angle subtended by the arc $\widehat{AB}$ at the center O of the circle is a radian. It is to be noted that a complete revolution makes two straight angles and the circumference of the circle is $2 \pi r$ units. Thus, we have

$$\frac{2 \pi r}{r} = \frac{2 Straight\ Angles}{1 Radian}$$

$\rightarrow$ 1 Straight Angle = $\pi ^c$

It is customary to denote the angles in the circular measure by the small letters $\alpha, \beta, \gamma$,… of the Greek Alphabet.

The angles in these three systems of measurement can be converted from one into another by the following relationship:

1 Straight Angle = $180^o$ = $200^g$ = $\pi ^c$

It should be noted that

1 Right Angle =$90^o$ = $100^g$ =

$\frac{\pi ^c}{2}$ Notation:If $\angle$AOB is an angle, it is customary to denote its measure by m $\angle$AOB.

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