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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.

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 complete rotation.

The measure of the angle traced during one complete rotation in anticlockwise direction is 360°. Similarly, the measure of the angle traced during two complete rotations in anticlockwise direction is just double i.e., 720° and so on.

Writing the measure of an angle:

Observe the following angles.

It can be seen that ∠POQ is a 45° angle, or we can say that ∠POQ measures 45°.

Mathematically, it is denoted as m∠POQ = 45°.

Similarly, m∠XOY = 90°, and m∠MON = 120°.

This is how we denote the measure of an angle.

Congruent angles:

Two angles are said to be congruent if their measures are equal.

From the above figures, it can be observed that ∠POQ = 45° = ∠BAC.

Thus, ∠POQ ∠BAC.

Properties of congruent angles:

(i) Reflexivity: Every angle is congruent to itself i.e., ∠PQR ∠PQR.

(ii) Symmetry: If ∠PQR ∠ABC, then ∠ABC ∠PQR.

(iii) Transitivity: If ∠PQR ∠ABC, and∠ABC ∠XYZ, then ∠PQR ∠XYZ.

Inequality of angles:

Out of two angles, the angle with greater measure is said to be greater than the angle with smaller measure.

In the above figures, we have

∠POQ = a° and ∠CAB = b°

If a >b, then ∠POQ > ∠CAB

Now, let us discuss some examples based on the concept of angle.

Example 1:

Name the angles marked in each of the following figures. Name the vertices and arms associated with these angles.

(i)

(ii)

(iii)

Solution:

(i) The angles marked in the figure are ∠QPR, ∠RPS, and ∠SPT.

The vertices and arms of each of these angles are listed as:

Angle

Vertex

Arms

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