PARALLEL LINES

Many elements in buildings – beams, pillars, windows, doors, window bars, flooring tiles – incorporate parallel lines. Line dividers on roads, railway lines, power lines are all examples of parallel lines. Parallel Lines assume a lot of importance when marking out roads or pedestrian crossings, sports courts, athletic tracks and airport runways.

Visually, we are so used to parallel-ness that most of us find it uncomfortable if we see a tube light not aligned parallel to the ceiling, or a tilted picture frame.

While we see them in everyday objects, we must ask, do they have any other importance?

They lie at the centre of many properties involving geometry. Drawing a transversal across pairs of parallel lines creates angles that have special properties.

In this pullout, we explore parallel lines and transversal properties. Solving any geometric problem requires knowledge as well as developing a geometric eye. Hence, in the teaching of geometry, we need to develop certain skills in children that will help to open a geometric eye. How does this geometric eye develop?

Over a period of 30 years, I have met students who were exposed to plenty of activities involving visual posers and who had a chance to play with Jigsaw puzzles, spot the hidden figure problems, spot visual patterns and so on in their primary school years. I have also met several who learnt geometry only in a limited classroom situation. I began to notice that the manner and skill with which these two groups of students approached a problem were quite different. The students who had greater exposure to visual challenges and greater contact with activities involving shapes had a higher ability to visualise and unpack the problem. While one cannot draw absolute conclusions about the kind of experiences that help to develop visual dexterity, I feel quite confident about this: exposing students to visual challenges does have a beneficial effect. Also, an intelligent guess at what one is looking for and having a sound knowledge of the geometric concepts aids the process.

Geometric problems require students to spot a specific object or a relationship. For example, an X (vertically opposite angle); a pair of adjacent supplementary angles; a pair of lines that are perpendicular to each other; or a pair of similar triangles; and all these in a figure which has many crisscrossing lines, angles and triangles. At times, we may have to turn the figure around to spot some of these things. There is a need to focus on relevant information and to ignore the rest of the data. In some ways, one has to turn a blind eye to irrelevant data.

How do we develop this geometric eye? What skills do we need to emphasise? I list a few here.

• Spot hidden geometric shapes.

• Spot right angles and straight angles.

• Spot pairs of equal line segments, perpendicular lines, and parallel lines.

• Spot pairs of shapes that are rotated relative to one another.

• Look at the same shape through different orientations: top-down, left-right.

• Find common line segments or angles in intersecting shapes.

• Spot shapes that are reflections and rotations of each other. Spot symmetries. Spot patterns.

• Hide some features of a diagram and highlight some others, using one’s fingers or actual highlighting.

 ACTIVITY 1

Objective: Warm up visual exercises of a general nature

Most of the exercises here are self-explanatory. They are all aimed at increasing observational skills. These include applying logic, recognising symmetry, visualising rotations and getting geometric ideas.

Jigsaw Puzzle: Draw an outline of the square, cut the shapes and ask the children to reassemble the pieces in the square.

 Complete the reflection.

Here are some rotations and reflections of F. Make some rotations and reflections of P.

Figure 3

How many rectangles do you see in this figure?

Figure 4

Describe the relative positions of squares A and B.

 

ACTIVITY 2

Objective: Warm-up activities to review necessary geometric facts

Complementary angles and supplementary angles property. 

Angles on a straight line add up to 180 degrees.

Opposite angles formed by two intersecting lines are equal.

Sum of the angles at a point is 360 degrees.

 I

f X = 20 degrees and Y is its supplement, what is Y – X?

If L = 50 degrees, M and L are complements, and N is a supplement of M, what is N?

What are a,b,c and d? (See Figure 8.)

ACTIVITY 3 Objective: Introduction to the concept of parallel lines, definition 

Show the students these pictures and ask them to comment on what they notice in them. 

What is the common thing that they see in all these pictures?

Ask them whether these lines will ever meet. Help them to come up with a definition for parallel lines in their own words initially. Give them the definition only after that.

Parallel lines are a pair of lines that lie on the same plane, and do not meet however far we extend them beyond both ends.

It is important that the lines should lie on the same plane. A line drawn on a table and a line drawn on the board may never meet but that does not make them parallel.

 


 

ACTIVITY 4

Objective: Spotting parallel lines 

 Note to the teacher: Drawings made on the board or on a ground are only approximations of parallel lines.

Let students look at various objects in the class to spot parallel lines in them.

Ex. Notebook edges, blackboard

Point out that when there are two sets of parallel lines, we use double arrows for the second set as shown in the picture.

How many sets of parallel lines do you see in this picture?