Take a piece of square paper and fold it in different ways. 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?
In this chapter, we will explore the relationship between lines on a plane surface. The table top, your piece of paper, the blackboard, and the bulletin board are all examples of plane surfaces.
Let us observe a pair of lines that meet each other. You will notice that they meet at a point. When a pair of lines meet each other at a point on a plane surface, we say that the lines intersect each other. Let us observe what happens when two lines intersect.
In Fig. 5.2, where line $l$ intersects line $m$, we can see that four angles are formed.
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.
We know that $\angle a$ and $\angle b$ together measure $180^\circ$, because when they are combined, they form a straight angle which measures $180^\circ$. So, if $\angle a$ is $120^\circ$, then $\angle b$ must be $60^\circ$.
Similarly, $\angle b$ and $\angle c$ together measure $180^\circ$. So, if $\angle b$ is $60^\circ$, then $\angle c$ must be $120^\circ$. And $\angle c$ and $\angle d$ together measure $180^\circ$. So, if $\angle c$ is $120^\circ$, then $\angle d$ must be $60^\circ$.
Therefore, in Fig. 5.2, $\angle a$ and $\angle c$ measure $120^\circ$, and $\angle b$ and $\angle d$ measure $60^\circ$.
When two lines intersect each other and form four angles, labelled $a, b, c$ and $d$, as in Fig. 5.2, then $\angle a$ and $\angle c$ are equal, and $\angle b$ and $\angle d$ are equal!
Check this for different measures of $\angle a$. Using these measurements, can you reason whether this property holds true for any measure of $\angle a$?
We can generalise our reasoning for Fig. 5.2, without assuming the values of $\angle a$.
Since straight angles measure $180^\circ$, we must have $\angle a + \angle b = \angle a + \angle d = 180^\circ$. Hence, $\angle b$ and $\angle d$ are always equal. Similarly, $\angle b + \angle a = \angle b + \angle c = 180^\circ$, so $\angle a$ and $\angle c$ must be equal.
Adjacent angles, like $\angle a$ and $\angle b$, formed by two lines intersecting each other, are called linear pairs. Linear pairs always add up to $180^\circ$.
Opposite angles, like $\angle b$ and $\angle d$, formed by two lines intersecting each other, are called vertically opposite angles. Vertically opposite angles are always equal to each other.
From the above reasoning, we conclude that whenever two lines intersect, vertically opposite angles are equal. Such a justification is called a proof in mathematics.
If two lines intersect and all four angles are equal, then each angle must be a right angle ($90^\circ$).
Perpendicular lines are a pair of lines which intersect each other at right angles ($90^\circ$). In Fig. 5.4, we can say that lines $l$ and $m$ are perpendicular to each other.
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.
For example, line segments FG and FH meet at the endpoint F at an angle $115.3^\circ$.
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? Here are some examples of lines we notice around us.
What is common to the lines in the pictures above? They do not seem likely to intersect each other. Such lines are called parallel lines.
Parallel lines are a pair of lines that lie on the same plane, and do not meet however far we extend them at both ends.
Name some parallel lines you can spot in your classroom.
Parallel lines are often used in artwork and shading.
Take a plain square sheet of paper (use a newspaper for this activity).
In mathematics, we use an arrow mark ($>$ ) to show that a set of lines is parallel. If there is more than one set of parallel lines (as in Fig. 5.9), the second set is shown with two arrow marks and so on. Perpendicular lines are marked with a square angle between them.
From previous exercises we observed that sometimes it is difficult to be sure whether two lines are parallel. To determine this we use the idea of transversals.
We saw what happens when two lines intersect in different ways. Let us explore what happens when one line intersects two different lines.
In Fig. 5.14, line $t$ intersects lines $l$ and $m$. $t$ is called a transversal. Notice that 8 angles are formed when a line crosses a pair of lines.
In Fig. 5.14, since $\angle 1$ and $\angle 3$ are vertically opposite angles, they are equal. Are there other pairs of vertically opposite angles? We can see that there are a total of four pairs of vertically opposite angles and in each pair, the angles are equal to each other.
Thus, when a transversal intersects two lines, it forms eight angles with a maximum of four distinct angle measures.
In Fig. 5.14, we notice that the transversal $t$ forms two sets of angles — one with line $l$ and another with line $m$. There are angles in the first set that correspond to angles in the second set based on their position. $\angle 1$ and $\angle 5$ are called corresponding angles. Similarly, $\angle 2$ and $\angle 6$, $\angle 3$ and $\angle 7$, $\angle 4$ and $\angle 8$ are the corresponding angles formed when transversal $t$ intersects lines $l$ and $m$.
Draw a pair of lines and a transversal such that they form two distinct angles.
When the corresponding angles formed by a transversal on a pair of lines are equal to each other, then the pair of lines are parallel to each other.
Suppose, we have a transversal intersecting two parallel lines. What can be said about the corresponding 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?
Corresponding angles formed by a transversal intersecting a pair of parallel lines are always 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.
Are you finding it hard to draw a transversal such that the corresponding angles are equal?
When a pair of lines are not parallel to each other, the corresponding angles formed by a transversal can never be equal to each other.
Can you draw a pair of parallel lines using a ruler and a set square? Fig. 5.21 shows how you can do it.
Draw a line $l$ with a scale. By sliding your set square you can make two lines perpendicular to line $l$.
Are these two lines parallel to each other? How are we sure that they are parallel to each other? What angles are formed between these lines and line $l$?
Since we used a set square, the angles measure $90^\circ$. The position of the lines is different, but they make the same angle with $l$. If line $l$ is seen as a transversal to the two new lines, then the corresponding angles measure $90^\circ$.
As we know these are corresponding angles and they are equal, we can be sure that the lines are parallel.
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?
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.
Let us try to do the same with paper folding. For a line $l$ (given as a crease), how do we make a line parallel to $l$ such that it passes through point A?
We know how to fold a piece of paper to get a line perpendicular to $l$. Now, try to fold a perpendicular to $l$ such that it passes through point A. Let us call this new crease $t$.
Now, fold a line perpendicular to $t$ passing through A again. Let us call this line $m$. The lines $l$ and $m$ are parallel to each other.
In Fig. 5.25, $\angle d$ is called the alternate angle of $\angle f$, and $\angle c$ is the alternate angle of $\angle e$.
You can find the alternate angle of a given angle, say $\angle f$, by first finding the corresponding angle of $\angle f$, which is $\angle b$ and then finding the vertically opposite angle of $\angle b$, which is $\angle d$.
In Fig. 5.25, if $\angle f$ is $120^\circ$ what is the measure of its alternate angle $\angle d$?
We can find the measure of $\angle d$ if we know $\angle b$ because they are vertically opposite angles. Remember, vertically opposite angles are equal.
What is the measure of $\angle b$? It is $120^\circ$ because it is the corresponding angle of $\angle f$.
So, $\angle d$ also measures $120^\circ$.
In fact, $\angle f = \angle b$ irrespective of the measure of $\angle f$. Why? Because $\angle b$ is the corresponding angle of $\angle f$.
Similarly, $\angle b = \angle d$ irrespective of the measure of $\angle b$. Why? Because $\angle d$ is the vertically opposite angle of $\angle b$. So, it must always be the case that $\angle f = \angle d$.
Using our understanding of corresponding angles without any measurements, we have justified that alternate angles are always equal.
Alternate angles formed by a transversal intersecting a pair of parallel lines are always equal to each other.
Solution: $\angle 6$ is $135^\circ$, so $\angle 2$ is also $135^\circ$, because it is the corresponding angle of $\angle 6$ and the lines $l$ and $m$ are parallel.
$\angle 8$ is $135^\circ$, because it is the vertically opposite angle of $\angle 6$. $\angle 4$ is $135^\circ$ because it is the corresponding angle of $\angle 8$.
$\angle 2$ is $135^\circ$ because is the vertically opposite angle of $\angle 4$. So, $\angle 2, \angle 4, \angle 6$, and $\angle 8$ are all $135^\circ$.
$\angle 5$ and $\angle 6$ are a linear pair, together they measure $180^\circ$. If $\angle 6$ is $135^\circ$, then $\angle 5 = 180^\circ - 135^\circ = 45^\circ$.
We can similarly find out that $\angle 1, \angle 3$, and $\angle 7$ measure $45^\circ$.
Solution: $\angle a$ is $120^\circ$, so $\angle b$ is $60^\circ$ because $\angle a$ and $\angle b$ form a linear pair. $\angle b$ is a corresponding angle of $\angle f$. If $l$ and $m$ are parallel, $\angle b$ should be equal to $\angle f$, however, they are not equal.
Therefore, lines $l$ and $m$ are not parallel to each other as the corresponding angles formed by the transversal $t$ are not equal to each other.
Solution: $\angle 3$ is $50^\circ$; therefore, $\angle 2$ is $130^\circ$, because $\angle 2$ and $\angle 3$ form a linear pair, and linear pairs always add up to $180^\circ$.
$\angle 2$ and $\angle 6$ are corresponding angles, and they need to be equal since lines $l$ and $m$ are parallel.
So, $\angle 6$ is $130^\circ$.
Angles $\angle 3$ and $\angle 6$ are called interior angles.
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. You will find that the sum of the interior angles on the same side of the transversal always add up to $180^\circ$.
Solution: Let us observe the parallel lines AB and CD. AD is a transversal of these two lines.
We know that the sum of the interior angles formed by a transversal on a pair of parallel lines adds up to $180^\circ$. So
$\angle ADC + \angle DAB = 180^\circ$
$60^\circ + \angle DAB = 180^\circ$.
So $\angle DAB = 120^\circ$.
Can we find $\angle CAB$ from this?
$\angle DAB = \angle DAC + \angle CAB$.
So $120^\circ = 65^\circ + \angle CAB$.
So $\angle CAB = 55^\circ$.
Let us observe the parallel line segments AD and BC. They are intersected by a transversal CD. So, $\angle ADC + \angle BCD = 180^\circ$, because they are interior angles on the same side of the transversal. Since $\angle ADC$ is given as $60^\circ$, $\angle BCD = 120^\circ$.
Similarly, we find $\angle ABC = 60^\circ$.
Therefore, in Fig. 5.29, $\angle CAB = 55^\circ$, $\angle ABC = 60^\circ$, and $\angle BCD = 120^\circ$.
There do not seem to be any parallel lines here. Or, are there?
What causes these illusions?