Class 6 – Geometry

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Geometry
  • Basic geometrical concepts
  • Angles
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Geometry
  • Parallel lines
  • Triangles
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Geometry
  • Quadrilaterals
  • Circles
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Introduction: Plane geometry, the branch of mathematics is concerned with the properties and measurements of points, lines, angles, planes, solids and surfaces and with their inter relationships, from the shape of snowflake to the orbit of a planet, from the construction level to the calculation for rockets into the outer space, the world bounced in geometric applications. Thus, the study of geometry is essential for fields of astronomy, agriculture, navigation, physics etc.

Point: A point is a mark of position. Usually, a fine dot marked with a sharp-edged pencil denoted on a plane paper, represents a point. We denote a point by a capital letter A, B, P, Q etc. In the adjoining figure. ‘A’ is a point. A point has neither length, breadth nor depth (or thickness).

Line Segment: Let A and B be two points on a paper. Then, the straight path from A to B is called the line segment AB, denoted by . We may also call it the line segment BA, denoted by .

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Thus, line segment is the same as line segment . The points A and B are called the end points of . The distance between the points A and B is called the length of .

Line: Geometrically, a line segment extended endlessly in both sides is called a line. A line segment AB extended on both sides and marked by arrow marks at the two ends represents a line, denoted by AB arrow or BA arrow. A line has no end points.

Two arrows in opposite directions of a line indicate that it extends indefinitely in both the directions. ∴ A line does not have a definite length.

Ray: Geometrically, a line segment extended endlessly in one direction is called a ray.

A line segment AB extended in the direction from A to B and marked by an arrow mark at B, represents a ray AB, denoted by AB single arrow.

A ray AB single arrow has one end point, namely A. This end point A of the ray AB single arrow is called its initial point. A ray has no definite length. Ray AB single arrow and ray BA single arrow are two different rays.

Opposite Rays: Two rays with the same initial point and extending indefinitely in opposite directions along the same line, are called opposite rays.

Differences between a Line Segment, a Line and a Ray:

Line Segment

Line

Ray

(i) A line segment has two end points.

A line has no end point.

A ray has one end point.

(ii) A line segment has a definite length

A line does not have a definite length.

A ray does not have a definite length.

(iii) A line segment of a given length can be drawn on a paper/plane.

A line cannot be drawn on a paper/plane. We can simply represent a line.

A ray cannot be drawn on a paper plane. We can simply represent a ray.

(iv) A line segment AB is represented by

A line AB is represented by AB arrow

A ray AB is represented by AB single arrow

(iv) Line segment is the same as line segment .

Line AB arrow is the same as line BA arrow.

Rays AB single arrow and BA single arrow are different.

Surface: A solid has a surface which may be curved or flat. For example, the surface of the ball is curved while the surface of the wall is flat.

Plane: A plane is a flat surface which extends indefinitely in all the directions. The surface of a smooth wall,the surface of the top of the table, the surface of a smooth black board, the surface of a sheet of paper, the surface of calm water in a pond are all examples of a portion of a plane.

Intersecting Lines: If there is a point P common to two lines l and m in the same plane,we say that the two lines intersect at the point P and this point P is called the point of intersection of the lines.

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Parallel lines: Twolines l and m in a plane which do not intersect even when produced are called parallel lines and we write l || m

The distance between two parallel lines always remains constant. Opposite edges of a ruler, opposite edges of the top of a table, railway track set c. are examples of parallel lines.

Incidence Properties of Lines in a Plane:

Property 1: An unlimited number of lines can be drawn passing through a given point.

Property 2: Only one line can pass through two distinct points in a plane.

Property 3: Two lines in a plane either intersect each other at exactly one point or parallel.

Concurrent lines: Three or more lines in a plane are said to be concurrent, if all of them pass through the same point. This common point is called the point of concurrence of the lines.

In the adjoining figure, the lines l, m, n, p all pass through the point P.So, all these lines are concurrent and P is the point of concurrence of these lines.

Collinear Points: Three or more points in a plane are said to be collinear if they all lie on the same line. This line is called the line of collinearity of these points.

Infig (i), all the four points A, B, C, D are collinear. In fig (ii), the points P, Q, R, S are non-collinear.

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Polygons: A Polygon is a 2-dimensional shape made of straight lines. Triangles and Rectangles are polygons.

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