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Basic Geometrical Ideas

Points, Line Segments, Lines, Rays, Planes and Space

Rohit and Mohit start walking from the same point in different directions. Let Rohit move towards Y and Mohit towards Z as shown below.

Here, we can say that the rays form an angle ZXY. These rays, i.e., are known as the arms of ∠ZXY. The point, i.e. X, that is common to these arms is called the vertex of ∠ZXY.

Let us consider the following ∠CAB with some points P, Q, and R.

Let us shade the different parts of the angle as shown below.

Here, we can see that the region of the angle shaded by yellow colour lies between the two arms of the angle. This region is called the interior region of the angle. It extends indefinitely, since the two arms also extends indefinitely. Every point in this region is said to lie in the interior of the angle. In this figure, point P lies in the interior of ∠CAB.

The region of the angle shaded pink lies outside the two arms of the angle. This region is called the exterior region of the angle. Like the interior region of an angle, the exterior region also extends indefinitely. Every point in this region is said to lie in the exterior of the angle. In this figure, point R lies in the exterior of ∠CAB.

The boundary of ∠CAB is formed by its arms. These arms are called the boundary of the angle. Every point lying on the arms is said to lie on the boundary of the angle, or simply, on the angle. In this figure, points A, B, C, and Q lie on the angle.

Using this concept, we can say that an angle has three regions. They are interior region, exterior region, and boundary region. Using this idea, we can easily identify whether a point lies inside, outside, or on the given angle. Let us discuss one more example to understand the concept better.

Example 1:

In the figure given below, name the point or points that lie

(i) in the interior of the angle

(ii) in the exterior of the angle

(iii) on the angle

Solution:

(i) Points B and E lie in the interior of the angle.

(ii) Points C and D lie in the exterior of the angle.

(iii) Points A, F, X, Y, and Z lie on the angle.

Look at the following figures.

(i)

(ii)

(iii)

Each of these shapes is an example of a curve. In fact, any shape that we draw is a curve. We can define a curve as follows.

Any figure drawn on a paper is known as a curve. A curve may or may not be straight.

Note: In real life, we do not consider straight lines as curves. However, in mathematics, straight lines are also considered as curves.

Can we find any difference among the three curves that we discussed in the beginning?

Curves (i) and (iii) do not intersect themselves, while curve (ii) does. Also, curves (i) and (iii) are not closed figures, while curve (ii) is a closed figure. On the basis of these observations, we classify curves as follows.

Simple curves Closed Curves Open curves

Let us discuss each of these with the help of the following video.

Let us discuss some more examples based on classification of curves.

Example 1:

Classify each of the following curves as open or closed.

(a)

(b)

(c)

(d)

Solution:

(a) Since no end points can be seen in the curve, it is an example of a closed curve.

(b) Since the two end points of the curve can be seen, it is an example of an open curve.

(c) Since the two end points of the curve can be seen, it is an example of an open curve.

(d) Since no end points can be seen in the curve, it is an example of a closed curve.

Example 2:

State whether each of the following curves is simple or not.

(a)

(b)

Solution:

(a) Since the curve does not cross itself, it is a simple curve.

(b) Since the curve crosses itself at one point, it is not a simple curve.

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.

Watch this video to learn more about angles.

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 r…

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