Multiple Lines & Shapes 2

Let us summarise the concepts that can be understood from the activities in the previous chapter.

School Geometry Curriculum

In these 2 activities, students formed these patterns with a little thought and without much help from the teacher. This is evidence that these patterns arise in our minds intuitively.

Geometers of ancient Greece must also have arrived at these patterns through a similar thought process. That is why the initial theorems in Plane Geometry are about points, lines, parallel lines, intersecting lines, angles and triangles.

It would be an interesting exercise to think of situations where some of the statements made below may not be true!

Points & Lines

In geometry ‘line’ means only a straight line. All the statements made below apply only to (straight) lines. Exceptions would be mentioned.

An infinite number of (curved or straight) lines can pass through the same point.

Through 2 given points only ONE (straight) line can be drawn.

Two (straight) lines can intersect only at ONE point.

All points through which a single (straight) line can be drawn are called collinear. Conversely all points on a (straight) line are collinear.

Parallel Lines - We will study parallel lines in a separate chapter. But as an introduction, let students try the following activity.

Take a piece of paper and cut the outer edge so that it does not contain any straight lines. They can imagine it as the shape of an amoeba! Fold 2 lines which are parallel to each other. They should also be able to prove logically that the 2 lines are parallel; mere visual appearance is not sufficient. We will see the solution in the chapter 22.18 on parallel lines.

Intersecting lines & Angles

Intersecting lines can form open or closed figures. They also form various angles and pairs of angles. We will explore the idea of angles in subsequent chapters.

Open & Closed Figures

Two kinds of figures can be formed using lines; Open & Closed figures. We can think of a closed figure as one which divides the plane surface into two or more distinct areas. Any point on the surface can be thought of as either being inside or outside the closed figure. An easier way is to remember it like the compound around a house with the gate closed.

In contrast, an open figure is one which does not divide the plane surface into two distinct area. We can even think of an open figure as just a curved or zig zag line. It can be thought of as a compound with the gate open.

Closed figures can be simple or not-simple. Any polygon is a simple closed figure. A figure like that of 8 is a not-simple closed curve, since it divides the surface into more than 2 distinct areas.

Sides of a Closed Figure

A minimum of three (straight) lines is required to form a closed figure. This closed figure formed with 3 lines is called a Triangle.

Closed figures can also be formed with more than 3 lines. They are called polygons.

However, with curved (lines) a closed figure can be formed with just one or two lines. The most familiar closed figure formed with just one curved line is the circle. The outline of the human eye is a closed figure formed with just 2 curved (lines).

Before introduction of geometry as a formal subject, students always think of simple shapes in terms of their names; triangle, square, rectangle, circle etc. When these shapes are introduced geometrically it would be useful to bring to their attention the number of sides which make up these geometrical shapes.

It would be a good activity to ask students to make 3,4,5 or 6 sided figures which are closed as well as open, using both straight lines & curved lines.

Line Segments

A line segment can be imagined as a line joining 2 given points. It has a fixed magnitude defined by the distance between the 2 points. If the two points are referred to as A & E, the line segment is referred to as AE and its magnitude is represented as ĀĒ. If this magnitude is measured in cm and hence, we could say ĀĒ = 6 cm.

We can think of a triangle as being bound by 3-line segments (or sides) each with a certain magnitude. The magnitudes of these sides are related by certain limitations expressed as theorems. One theorem is that the sum of any two sides would be greater than the third side.

This theorem can be generalized for any polygon. Any side of a polygon should be less than the sum of all the other sides.