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

We will perform a paper folding activity to understand lines.

Step 1 — Describing edges of a square sheet

Let us take a plain square sheet of paper. We look at its edges. The edges opposite to each other never meet. They stay the same distance apart. So, they are parallel.

The edges next to each other meet at a corner. They form a square corner. This is a right angle. So, they are perpendicular.

Step 2 — First horizontal fold

We fold the sheet horizontally in half. This creates a new line. We now have the top edge, the bottom edge, and this new fold line. These three lines are all horizontal. They are all parallel to each other. The new fold line is horizontal. The vertical sides of the paper are straight up and down. Horizontal lines and vertical lines form right angles. So, the new fold line is perpendicular to the vertical sides.

Step 3 — Second horizontal fold

We make one more horizontal fold. We fold the already folded sheet in half again. When we unfold the paper, we will see more lines. We have the two original edges. We also have three fold lines. All these lines are horizontal. They are all parallel to each other. So, we see 5 parallel lines in total.

Step 4 — Third horizontal fold and pattern

Let us make one more horizontal fold. This is a third horizontal fold. We fold the paper in half again. When we unfold it, we will see even more lines. We will have the two original edges. We will also have seven fold lines. So, we will get 9 parallel lines. There is a pattern here. Each time we fold, the number of sections doubles. The number of fold lines is one less than the number of sections. The pattern extends further. If we fold again, we will get more parallel lines.

Step 5 — Vertical fold

We take a fresh square sheet. We make a vertical fold in it. This new line goes up and down. The previous horizontal lines go side to side. Vertical lines and horizontal lines meet at right angles. So, this new vertical line is perpendicular to the horizontal lines.

Step 6 — Diagonal fold

We fold the sheet along a diagonal. This creates a diagonal fold line. We can find another fold that is parallel to this diagonal line. For example, we can fold a corner of the paper. We fold it so its edge lies on the diagonal line. This new fold line will be parallel to the diagonal. So, yes, it is possible.

Answer

(i) They are parallel to each other. (ii) The adjacent edges are perpendicular to each other. (iii) The two original top/bottom edges and the new horizontal fold. It is perpendicular to the vertical sides. (iv) 5 parallel lines (the two original edges plus the three horizontal folds). (v) Doing it once more (a third horizontal fold) would result in 9 parallel lines (2 original edges + 7 folds). The pattern relates to powers of 2; after each fold, the number of sections doubles, and the number of fold lines increases. The pattern extends further. (vi) This new vertical line is perpendicular to the previous horizontal lines. (vii) This requires performing the activity. It is possible to create such a fold (e.g., by folding an edge onto the diagonal or folding the paper in half parallel to the first diagonal fold).

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.