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Basic Geometrical Ideas

We have seen lamp posts and trees on the street side. These lamp posts and trees stand straight on the road as shown in the figure below.

What do we observe in the above figures?

We observe that there is a small curve denoted by A between the tree and the ground. Similarly, in the case of lamp post, a curve is denoted by B. These curves are known as angles. We say that the tree and the lamp post are making angles A and B respectively with the ground.

In some cases, it is difficult to specify an angle by its vertex. For example, in the figure given below, ∠A may denote any of the angles among ∠BAC, ∠CAD, or ∠BAD.

Therefore, it is desirable to represent an angle by three letters (which make the angle) and not just by the vertex letter.

Angles in terms of rotation:

An angle is obtained when a ray is rotated about its end point. The ray can be rotated in two ways such as clockwise (direction in which the hands of a clock move) and anticlockwise (direction opposite to clockwise direction).

When a ray is rotated clockwise, the obtained angle is regarded as negative while the ray is rotated anticlockwise, the angle obtained is regarded as positive.

Observe the following figures.

In figure (i), the angle is obtained by rotating the initial arm clockwise, thus this angle is negative. On the other hand, the angle obtained in figure (ii) is positive as it is obtained by rotating the initial arm anticlockwise.

The amount of rotation of the ray from its initial position to terminal position is known as the measure of the angle.

In figures (i) and (ii), the angles are not same, even if their measures are equal because one of them is positive and the other is negative.

Directed angle:

When a ray rotates about its origin point to occupy the position of another ray having the same origin point, it forms an angle which is known as the directed angle. Name of the directed angle is written according to the direction of rotation of initial arm.

For directed angle, rays can be represented in the form of ordered pair as (ray acting as initial arm, ray acting as terminal arm).

If we have two rays OP and OQ, then there can be two possibilities for directed angle. These are as follows:

(i) Ray OP is initial arm and ray OQ is terminal arm:

In this case, we get the ordered pair as ray OP, and ray OQ, which represents that the directed angle is obtained by the rotation of ray OP to occupy the position of ray OQ. Thus, obtained angle is called directed angle POQ.

Directed angle POQ is denoted as POQ.

(ii) Ray OQ is initial arm and ray OP is terminal arm:

In this case, we get the ordered pair as ray OQ, and ray OP, which represents that the directed angle is obtained by the rotation of ray OQ to occupy the position of ray OP. Thus, obtained angle is called directed angle QOP.

Directed angle QOP is denoted as QOP.

So, it can easily be concluded that the directed angle POQ and directed angle QOP are different.

i.e., POQ ≠ QOP

Positive and negative angles:

Angle obtained on anticlockwise rotation of initial arm is regarded as positive angle.
For example, POQ in the above shown figure is positive angle.

Angle obtained on clockwise rotation of initial arm is regarded as negative angle.
For example, QOP in the above shown figure is negative angle.

One complete rotation:

If the initial ray OP is rotated about its end point O in anticlockwise direction such that it comes back to the position OP again for the first time, then it is said that the ray OP has formed one completâ€¦

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