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?

Parallel and Intersecting Lines | A

Question 4

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?

Solution

Understand the Question

Parallel Lines: Lines in the same plane that never meet, no matter how far they are extended, remaining at a constant distance from each other.
Perpendicular Lines: Lines that intersect each other at a right angle ( $90^\circ$ ).
In a square sheet, opposite edges are parallel, while adjacent edges meet at right angles and are perpendicular.
Repeatedly folding a sheet along parallel directions doubles the number of sections and creates parallel creases.

(i) How would you describe the opposite edges of the sheet? They are _______________________ to each other.

Answer

(i) parallel

(ii) 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.

Answer

(ii) perpendicular

(iii) 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?

Step 1 · Analyze the First Horizontal Fold

Question diagram * Folding horizontally in half creates $1$ new horizontal crease.

Including the top and bottom horizontal edges, there are $3$ horizontal parallel lines in total.
The horizontal fold line meets the vertical edges at right angles ( $90^\circ$ ), so it is perpendicular to the vertical sides.

Answer

(iii) $3$ parallel lines; it is perpendicular to the vertical sides.

(iv) Make one more horizontal fold in the folded sheet. How many parallel lines do you see now?

Step 1 · Count Parallel Lines After Second Fold

Folding the sheet horizontally in half again creates $3$ horizontal creases.

$\text{Total parallel lines} = 2 \text{ (edges)} + 3 \text{ (folds)} = 5 \text{ lines}$

Answer

(iv) $5$ parallel lines

(v) 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.

Step 1 · Establish the Pattern for Subsequent Folds

Each successive fold doubles the number of equal horizontal sections.

For $n$ folds, the number of sections is $2^n$ , which gives $2^n - 1$ fold lines: $\text{Total parallel lines} = \text{edges} + \text{folds} = 2 + (2^n - 1) = 2^n + 1$

For a third fold ( $n = 3$ ): $\text{Total parallel lines} = 2^3 + 1 = 8 + 1 = 9$

This pattern continues for further folds (e.g., for $n = 4$ , there are $2^4 + 1 = 17$ parallel lines).

Answer

(v) $9$ parallel lines. Yes, after $n$ folds, there are $2^n + 1$ parallel lines, and the pattern extends for further folds.

(vi) Make a vertical fold in the square sheet. This new vertical line is _______________________ to the previous horizontal lines.

Answer

(vi) perpendicular

(vii) Fold the sheet along a diagonal. Can you find a fold that creates a line parallel to the diagonal line?

Step 1 · Create a Parallel Diagonal Crease

Fold a corner vertex toward the main diagonal crease such that the new crease does not intersect the diagonal and remains equidistant from it.

This new fold line is parallel to the original diagonal.

Answer

(vii) Yes, it is possible to create a fold parallel to the diagonal line.

Common Mistakes

Forgetting Edges: Only counting the internal fold lines and forgetting to include the original outer boundary edges when counting total parallel lines.
Confusing Parallel and Perpendicular: Describing horizontal lines relative to vertical lines as parallel rather than perpendicular ( $90^\circ$ intersection).
Linear vs Exponential Growth: Assuming the number of parallel lines increases by a constant addition rather than doubling the number of sections ( $2^n + 1$ total lines after $n$ folds).

More questions in A

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?

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.