Fundamental Laws of Arithmetic

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Let us now look at other types of angles and their measures. All of them are defined with respect to a right or straight angles.

Acute Angle

All angles less than a Right Angle are called Acute angles. Hence the measure of an Acute Angle is &lt; 90&deg;.

Obtuse Angle

All angles bigger than a right angle, but smaller than a Straight angle, are called Obtuse angles. Hence the measure of an Obtuse Angle is &gt; 90&deg;.

Reflex Angle

An angle which is bigger than a Straight Angle but smaller than a complete angle is called a Reflex Angle. Hence the measure of Reflex Angle is &gt;180&deg;. There are very few applications of Reflex angles in daily life possibly because every reflex angle comes with a built-in Acute angle!

If the above angles are arranged in ascending order of their magnitude it would be - Zero Angle, Acute Angle, Right Angle, Obtuse Angle, Straight Angle, Reflex Angle &amp; Complete Angle.

Summary of Relations &amp; Measures

The relations between various angles are summarized in a table below.

Measurement of Angles

Angles are measured using a Protractor, which is divided into 360 parts, each of which represents a degree. The symbol for degrees is “&deg;”. For convenience of measuring in both directions, the markings are done both clockwise and anti-clockwise. Hence there are 2 numbers against each mark. For example, one of the arms of an angle could be considered as at 30&deg; in one direction and 150&deg; in the opposite direction. The actual measure depends on which angle is to be measured. This can be very confusing to students who do not have a clear idea of the rotational aspect of an angle.

Visual Perception

Hence, before learning to measure angles with a protractor, students should get a lot of practice in identifying the type of the angle and estimating the measure of angles.

For this a lot of cut-outs can be made from cardboard of different angles. The cut-outs should be made from a circular sheet, so that one of the sides of the angle is an arc and the other 2 sides, representing the radius of the circle are equal. This will avoid the confusion between an angle and a triangle.

I am also sharing a very nice visual on approximating important angles like 30&deg;, 45&deg;, 60&deg; & 90&deg; with our fingers.



Using Set Squares

The pair of Set Squares in a geometry box can also be used to work out the range of an angle without actually measuring it. The set squares are made with definite angles – one with angles 30-60 &amp; 90 and the other with 45-45-90. Hence the angles 30, 45, 60 &amp; 90 are available in concrete form.

The approximate measure of an angle can be estimated by using the various set square angles.

In the next chapter, after studying addition &amp; subtraction of angles, we will see that the Set Squares can be used to form angles which are either sum or difference of the angles available on both the Set Squares.

Why is the day divided into 24 hours?

We are not sure as to who invented the idea of dividing a day into 24 hours. But there is an interesting story connecting measurement of angles with the number of hours in a day!

We have seen why the ancients decided that a complete angle should be divided into 360&deg;. They were also familiar with the construction of angles measuring 30, 45, 60 &amp; 90&deg; using just a compass and a straight edge. Using the idea of differences, they could also construct angles measuring 15&deg;.

They possibly divided the day into 24 hours, because, an hour is the time the Sun takes to move through 15&deg;! (24 X 15 = 360)

They took 15&deg; as a measure possibly because bisecting an angle of 15&deg; led them to 7 &frac12;&deg;. Their number systems must have made it very difficult for them to handle fractions!

< 21.1 Properties & Relations of Operations | Topic Index | 21.3 Order of Operations >