Basic Geometrical Ideas: Geometry Basics, Polygons, Lines

Geometry is the branch of studying the position, shape, size and other properties of various figures. It has a long and vibrant history. Geo means earth, and metron implies measurement. Basic geometrical ideas will help students learn more difficult topics in geometry.

According to historians, the geometrical idea was developed in ancient times due to art, architecture, and measurement. The construction of palaces, temples, lakes, dams, cities, art, and architecture brought this idea to read and learn more about geometrical figures. In this article, we will cover the basics of shapes around us, i.e., we will read about the basics of figures in geometry.

What are Basic Geometrical Ideas?

You might have observed and used different objects like boxes, tables, books, tiffin box you carry to school, the ball with which you play, the top of your desk, the shape of the blackboard on which your teacher writes, the ruler which you use, and so on. All such objects have different shapes. Geometry deals with studying the position, shape, size, and other properties of such figures or objects.

Basic geometrical ideas give us an overview of the basic terms used in geometry. Let us learn about those basics terms in detail.

1. Point: A point shows a definite position or location which cannot be moved, i.e., a point is a mark of place. A point has neither length nor width nor thickness and thus occupies no space. A point is represented by a dot made with a sharp pencil on a sheet of paper. Points are usually named using single capital letters. We can represent points by the head or tip of a drawing pin, the tip of a sharp needle, etc.

2. Line: A collection of points makes a line. Thus, a line has the only length. It has neither width nor thickness. A line extends endlessly or indefinitely in both directions. Therefore, we always draw a line with arrows at both ends. 

A line has no endpoints. So, it has no beginning or no end, and thus, one can not measure its length. 

Take a rectangular piece of paper, fold it precisely at its middle, and press the two parts together. On unfolding it, we will get the straight crease. The crease so obtained is the physical example of a part of a line.

Another physical example of a part of a line is a tight straight thread.

We can denote a line in two ways as follows:

1. Using a single small letter, for example, line \(l\) or line \(p\).
2. Using any two points marked on it, for example, line \(A B\), or \(\mathop {AB}\limits^ \leftrightarrow .\)

3. Ray: A ray is a part of a line that has one endpoint and can be extended in the other direction indefinitely. In simple words, a ray is a straight line that starts from a given fixed point and moves in the same direction. The fixed endpoint is called the initial point of the ray. We cannot measure the length of a ray. We can write ray \(A B\) as \(\mathop {AB}\limits^ \leftrightarrow .\) A line consists of an infinite number of rays.

4. Line Segment: A part of a line is called a line segment. It has \(2\) endpoints and has a definite length, which cannot be increased or decreased. We can write a line segment \(A B\) as \(\overline{A B}\). A line segment is a part of a line as well as of a ray. Also, between any two points, we can draw only one line segment.

5. Plane: The surface of any solid can be flat or curved. The surface of a table is flat, but the surface of the ball is curved. A flat surface gives an idea of a plane. A plane is a flat \(2-D\) surface that can be extended endlessly in all directions. A plane is a collection of an infinite number of points. It also contains an infinite number of lines, line segments,
and rays.

6. Collinear and Non-Collinear Points: If \(3\) or more points lie on the same straight line, then the points are called collinear points. At \(6\) ‘o clock, the centre of the clock and the endpoints of the short and long hands form the collinear points. In the given figure, points \(A, B\), and \(C\) lie on the same line.

When \(3\) points do not lie on the same line, we call them non-collinear points. In the below-given figure, points \(D, E\), and \(F\) does not lie on the line \(\mathop {AC}\limits^ \leftrightarrow .\)

Curves

A curve is a continuous line without any sharp and steep turns. The other way to define a curve is that a line that changes its direction at least once. Have you ever taken a piece of paper and just doodled it?? The pictures that are the results of your doodling are known as curves.

We can draw some of these drawings without lifting the pencil from the paper and without a ruler. A curve in everyday usage means not straight. In the above-given figure, some curves cross themselves, and some do not.

Thus, if a curve does not cross itself, then it is called a simple curve.

Polygons

A polygon is a flat shape made up of line segments connected end to end, forming a closed figure, or in short, a polygon is a closed plane figure bounded by straight line segments.

The straight-line segments which make up a polygon are called the sides of a polygon, and endpoints of the line segments are called vertices of the polygon.

We name polygons according to the number of sides they contain. For example, if the polygon has three sides, it is called a triangle. If the polygon has five sides, it is called a pentagon. If the polygon has six sides, it is called a hexagon and so on.

Angles

Two different rays starting from the same fixed point form an angle. In the figure given below, two different rays \(QP\) and \(QR\) start from the same fixed point \(Q\) to form angle \(PQR\). The point \(Q\), which is common to both the rays, is called the vertex of the angle \(PQR\), whereas the ray \(QP\) and \(QR\) are called the sides or arms of the angle \(PQR\). An angle is represented by \(3\) capital letters taken so that the letter in the middle is always the vertex of the angle.

The interior of an angle is the region that lies within an angle. In other words, it is the region bounded by the arms of an angle, whereas the exterior of an angle is the region that lies outside the angle.

The image given below will give us a better look and understanding of an angle’s interior and exterior points.

Triangles

A triangle is a plane closed figure bounded by \(3\) line segments. In the figure given below, the lien segments, \(AB, BC\), and \(CA\) form the \(\triangle ABC\).

The \(3\) line segments \(AB, BC\) and \(CA\) are the sides of the triangle. Vertex of a triangle is a point where any two of its sides meet.

Solved Examples

Q.1. How many lines can be drawn, passing through:
(a) given point      
(b) Two given fixed points                
(c) \(3\) non-collinear points
Ans: A collection of points makes a line. Thus, a line has the only length.
a) A given point: Through a given point, we can draw infinite lines.
b) Two given fixed points: Through two given fixed points, we can draw only one line.       
c) \(3\) non-collinear points: Through \(3\) non-collinear points, we can draw zero lines.

Q.2. Give one example, from your surrounding, for each of the following:
(a) Plane surfaces
(b) Curved surfaces.
Ans: a) Plane surface: Floor
b) Curved surface: Football

Q.3. Name the points:
a) In the exterior of \(\angle P Q R\)
b) In the interior of \(\angle P Q R\)

Ans: The interior of an angle is the region that lies within an angle. In other words, it is the region bounded by the arms of an angle, whereas the exterior of an angle is the region that lies outside the angle.
Thus, the points in the exterior of \(\angle P Q R\) are \(m, d, n, s\) and \(t\), whereas the points in the interior of \(\angle P Q R\) are \(a, b\) and \(x\).

Q.4. In the below-given figure, write the name of the vertex, the names of the arms, and the name of the angle.

Ans: Arms \(=A B\) and \(B C\), vertex \(=O\) and angle \(=\angle A B C\)

Q.5. Under what condition will two straight lines in the same plane have no point in common?
Ans: A pair of parallel lines lying on the same plane will have no point in common.

Summary

In this article, we learned the basics used in geometry. We learned about them through their definition and with the help of examples like point, ray, line, line segments, angles, curves, etc. We also learnt about angles, polygons, triangles, and curves. We also studied some solved examples to understand the concept in a better way.

FAQs

Q.1. What are basic geometrical ideas?
Ans: The basic geometrical ideas give us an overview of the basic terms and shapes in geometry. It details what a point, line, line segment, ray, curve, etc. are.

Q.2. What is basic geometry?
Ans: You might have observed and used different objects like boxes, tables, books, tiffin box you carry to school, the call with which you play, the top of your desk, the shape of the blackboard on which your teacher writes, the ruler which you use, and so on. All such objects have different shapes. Geometry is studying the position, shape, size, and other properties of various such figures.

Q.3. What are the basic shapes of geometry?
Ans: Triangle, circle, oval, rhombus, rectangle, square, trapezoid, pentagon, hexagon, and octagon are the basic geometry shapes.

Q.4. What is a line ray?
Ans: A ray is a part of a line that has one endpoint and can be extended in the other direction indefinitely. In simple words, a ray is a straight line that starts from a given fixed point and moves in the same direction. The fixed endpoint is called the initial point of the ray.

Q.5. What is a curve in maths?
Ans: A curve is a continuous line without any sharp and steep turns. The other way to define a curve is that a line that changes its direction at least once. Have you ever taken a piece of paper and just doodled it?? The pictures that are the results of your doodling are known as curves.

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