Parallel and Intersecting Lines

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.

List all the linear pairs and vertically opposite angles you observe in Fig. 5.3:

Draw some lines perpendicular to the lines given on the dot paper in Fig. 5.10.

In Fig. 5.11, mark the parallel lines using the notation given above (single arrow, double arrow etc.). Mark the angle between perpendicular lines with a square symbol.

(a) How did you spot the perpendicular lines?

(b) How did you spot the parallel lines?

In the dot paper following, draw different sets of parallel lines. The line segments can be of different lengths but should have dots as endpoints.

Using your sense of how parallel lines look, try to draw lines parallel to the line segments on this dot paper.

(a) Did you find it challenging to draw some of them? (b) Which ones? (c) How did you do it?

In Fig. 5.13, which line is parallel to line a — line b or line c? How do you decide this?

Can you draw a line parallel to $l$, that goes through point A? How will you do it with the tools from your geometry box? Describe your method.

Find the angles marked below.

Find the angle represented by $a$.

In the figures below, what angles do $x$ and $y$ stand for?

In Fig. 5.33, $\angle ABC = 45^\circ$ and $\angle IKJ = 78^\circ$. Find angles $\angle GEH$, $\angle HEF$, $\angle FED$

In Fig. 5.34, $AB$ is parallel to CD and CD is parallel to EF. Also, EA is perpendicular to AB. If $\angle\text{BEF} = 55^\circ$, find the values of $x$ and $y$.

What is the measure of angle $\angle\text{NOP}$ in Fig. 5.35?

[Hint: Draw lines parallel to LM and PQ through points N and O.]

Context: Let us observe what happens when two lines intersect.

Q. How many angles do they form?

Can two straight lines intersect at more than one point?

In Fig. 5.2, if $\angle a$ is $120^\circ$, can you figure out the measurements of $\angle b$, $\angle c$ and $\angle d$, without drawing and measuring them?

Context: When two lines intersect each other and form four angles, labelled $a$, $b$, $c$ and $d$, then $\angle a$ and $\angle c$ are equal, and $\angle b$ and $\angle d$ are equal.

Q. Is this always true for any pair of intersecting lines?

Can you draw a pair of intersecting lines such that all four angles are equal? Can you figure out what will be the measure of each angle?

Observe Fig. 5.5 and describe the way the line segments meet or cross each other in each case, with appropriate mathematical words (a point, an endpoint, the midpoint, meet, intersect) and the degree measure of each angle.

Are line segments ST and UV likely to meet if they are extended?

Are line segments OP and QR likely to meet if they are extended?

Name some parallel lines you can spot in your classroom.

Which pairs of lines appear to be parallel in Fig. 5.6 below?

Is it possible for all the eight angles to have different measurements? Why, why not?

What about five different angles — 6, 5, 4, 3 and 2?

Suppose, we have a transversal intersecting two parallel lines. What can be said about the corresponding angles?

Context: Activity 5 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.

Q. Are you finding it hard to draw a transversal such that the corresponding angles are equal?

Draw two more parallel lines using the long side of the set square as shown in Fig. 5.22. How do you know these two lines are parallel? Can you check if the corresponding angles are equal?

Why are lines $l$ and $m$ parallel to each other?

Is there a relation between $\angle 3$ and $\angle 6$? You could try to find the relationship by taking different values for $\angle 3$ and see what $\angle 6$ is. Once you find a relation, try to justify it or prove that this relation holds always.

There do not seem to be any parallel lines here. Or, are there?

What causes these illusions?