Parallel and Intersecting Lines | A

Question 1

Take a piece of square paper and fold it in different ways. Now, on the creases formed by the folds, draw lines using a pencil and a scale. You will notice different lines on the paper. Take any pair of lines and observe their relationship with each other. Do they meet? If they do not meet within the paper, do you think they would meet if they were extended beyond the paper?

Solution

Understand the Question

When paper is folded in different ways, the resulting crease lines show two main types of geometric relationships:
- Intersecting Lines: Lines that cross or meet at a single point, either on the paper itself or when extended beyond it.
- Parallel Lines: Lines that are always the same distance apart and never meet, no matter how far they are extended in either direction.

Step 1 · Observe Intersecting Lines

Diagram 1

Consider the two diagonal crease lines:

One line runs from the top-left to the bottom-right corner, and the other runs from the top-right to the bottom-left corner.
These two lines cross each other at a single point inside the paper.
Lines that cross at a point are called intersecting lines.

Step 2 · Observe Parallel Lines

Diagram 2

Consider the two vertical lines:

These lines run from the top edge to the bottom edge and remain an equal distance apart.
They do not meet within the paper and will never meet even if extended indefinitely.
Lines in a plane that never meet when extended are called parallel lines.

Answer

Yes, some pairs of lines meet within the paper (intersecting lines).

Lines that do not meet within the paper will meet if extended, unless they are equidistant.
Lines that remain at a constant distance apart will never meet, even when extended beyond the paper (parallel lines).

Common Mistakes

Assuming non-meeting lines on paper are always parallel: Lines that do not cross within the paper boundaries may still intersect if extended further, unless they maintain a constant distance apart.
Parallel Line Condition: Parallel lines must always stay the exact same perpendicular distance apart at all points.

More questions in A

Draw two lines on a plain sheet of paper so that they intersect. Measure the four angles formed with a protractor. Draw four such pairs of intersecting lines and measure the angles formed at the points of intersection.

What patterns do you observe among these angles?

Activity 2 Take a plain square sheet of paper (use a newspaper for this activity).

How would you describe the opposite edges of the sheet? They are _______________________ to each other.
How would you describe the adjacent edges of the sheet? The adjacent edges are _______________________ to each other. They meet at a point. They form right angles.
Fold the sheet horizontally in half. A new line is formed (see Fig. 5.7).
How many parallel lines do you see now? How does the new line segment relate to the vertical sides?
Make one more horizontal fold in the folded sheet. How many parallel lines do you see now?
What will happen if you do it once more? How many parallel lines will you get? Is there a pattern? Check if the pattern extends further, if you make another horizontal fold.
Make a vertical fold in the square sheet. This new vertical line is _______________________ to the previous horizontal lines.
Fold the sheet along a diagonal. Can you find a fold that creates a line parallel to the diagonal line?

Here is another activity for you to try.

Take a square sheet of paper, fold it in the middle and unfold it.
Fold the edges towards the centre line and unfold them.
Fold the top right and bottom left corners onto the creased line to create triangles. Refer to Fig. 5.8.
The triangles should not cross the crease lines.
Are $a$ , $b$ and $c$ parallel to $p$ , $q$ and $r$ respectively? Why or why not?

Draw a pair of lines and a transversal such that they form two distinct angles.

Fig. 5.19 has a pair of parallel lines $l$ and $m$ (what is the notation used in the figure to indicate they are parallel?). Line $t$ is the transversal across these two lines. $\angle a$ and $\angle b$ are corresponding angles. Take a tracing paper and trace $\angle a$ on it. Now place this tracing paper over $\angle b$ and see if the angles align exactly. You will observe that the angles match. Check the other corresponding angles in the figure using a protractor. Are all the corresponding angles equal to each other?

In Fig. 5.20, draw a transversal $t$ to the lines $l$ and $m$ such that one pair of corresponding angles is equal. You can measure the angles with a protractor.