NCERT Class 7 Ganita Prakash 1 Chapter 05 Parallel and Intersecting Lines PDF Download

Chapter 5: Parallel and Intersecting Lines

5.1 Across the Line

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?

[Figure 5.1: Square paper with fold creases creating intersecting lines, See in your textbook]

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.

How many angles do they form?

In Fig. 5.2, where line \( l \) intersects line \( m \), we can see that four angles are formed.

[Figure 5.2: Two intersecting lines l and m forming four angles a, b, c, and d, See in your textbook]

Can two straight lines intersect at more than one point?

Activity 1

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?

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

We know that \( \angle a \) and \( \angle b \) together measure 180°, because when they are combined, they form a straight angle which measures 180°. So, if \( \angle a \) is 120°, then \( \angle b \) must be 60°.

Similarly, \( \angle b \) and \( \angle c \) together measure 180°. So, if \( \angle b \) is 60°, then \( \angle c \) must be 120°. And \( \angle c \) and \( \angle d \) together measure 180°. So, if \( \angle c \) is 120°, then \( \angle d \) must be 60°.

Therefore, in Fig. 5.2, \( \angle a \) and \( \angle c \) measure 120°, and \( \angle b \) and \( \angle d \) measure 60°.

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!

Teacher's Note

When two lines intersect, you get four angles. The angles across from each other (like \( \angle a \) and \( \angle c \)) are always equal - these are called vertically opposite angles. The angles next to each other (like \( \angle a \) and \( \angle b \)) always add up to 180° - these are called linear pairs. These properties are true for any intersecting lines, no matter what the actual angle measures are.

Is this always true for any pair of intersecting lines?

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°, we must have \( \angle a + \angle b = \angle a + \angle d = 180° \). Hence, \( \angle b \) and \( \angle d \) are always equal. Similarly, \( \angle b + \angle a = \angle b + \angle c = 180° \), 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°.

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.

Figure it Out

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

[Figure 5.3: Two intersecting lines l and m with angles a, b, c, and d marked, See in your textbook]

Measurements and Geometry

You might have noticed that when you measure linear pairs, sometimes they may not add up to 180°. Or, when you measure vertically opposite angles they may be unequal sometimes. What are the reasons for this?

There could be different reasons:

• Measurement errors because of improper use of measuring instruments - in this case, a protractor
• Variation in the thickness of the lines drawn. The "ideal" line in geometry does not have any thickness! But it is not possible for us to draw lines without any thickness

In geometry, we create ideal versions of "lines" and other shapes we see around us, and analyse the relationships between them. For example, we know that the angle formed by a straight line is 180°. So, if another line divides this angle into two parts, both parts should add up to 180°. We arrive at this simply through reasoning and not by measurement. When we measure, it might not be exactly so, for the reasons mentioned above. Still the measurements come out very close to what we predict, because of which geometry finds widespread application in different disciplines such as physics, art, engineering and architecture.

Teacher's Note

In geometry, we use reasoning and proof to establish facts, not just measurement. Your protractor measurements might be slightly off because of how you hold it or because the lines have thickness, but the mathematical truth remains: linear pairs must equal 180° and vertically opposite angles must be equal. Always trust the logic and the proof, even when measurements seem slightly different.

5.2 Perpendicular 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?

Key Points

• When two lines intersect, they form four angles. Angles that are next to each other form a linear pair and always add up to 180°.
• Angles that are across from each other at an intersection are called vertically opposite angles, and they are always equal to each other.
• In geometry, we use logical reasoning and proofs to establish that relationships between lines and angles are always true, even though our measurements might not be perfectly exact.
• A line on a plane surface has no thickness, but when we draw lines on paper they do have thickness, which is why our measurements might differ slightly from the theoretical predictions.

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