Parallel and Intersecting Lines – Class 7 Maths Chapter 5 Notes & NCERT PDF

Class 7MathematicsChapter 5NCERT book: Ganita Prakash - Mathematics Class 7

Notes and a simple summary of Chapter 5 of the NCERT Class 7 Mathematics book, with the official chapter PDF, flashcards, an MCQ quiz and an AI tutor for your doubts.

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Chapter 5: Parallel and Intersecting Lines

Introduction

The chapter begins with an activity: fold a square sheet of paper in different ways and draw along the creases. Some pairs of lines meet; others never seem to. This chapter studies how lines on a plane surface (a table top, a sheet of paper, a blackboard) relate to each other.

Key Ideas

5.1 Across the Line: Intersecting Lines

When two lines meet at a point, they intersect. Two straight lines can meet at only one point, and they form four angles (called a, b, c, d in Fig. 5.2).

If ∠a = 120°, then ∠b = 60° (together they make a straight angle of 180°), ∠c = 120° and ∠d = 60°.

Linear pairs are adjacent angles formed by two intersecting lines; they always add up to 180°.

Vertically opposite angles are opposite angles formed by two intersecting lines; they are always equal.

Why vertically opposite angles are equal (a proof): ∠a + ∠b = 180° and ∠a + ∠d = 180°, so ∠b = ∠d. Similarly ∠a = ∠c. We reached this by reasoning, not measuring. When we measure, small errors can appear because of the protractor or because drawn lines have thickness, but an "ideal" geometric line has none.

5.2 Perpendicular Lines

If two intersecting lines make four equal angles, each is 360° ÷ 4 = 90°. Such lines are perpendicular.

5.3 Between Lines: 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.

Railway tracks and the lines of a ruled notebook are examples. The lines must lie on the same plane: a line on the table and one on the board may never meet, but they are not parallel.

5.4 Parallel and Perpendicular Lines in Paper Folding

Opposite edges of a square sheet are parallel; adjacent edges are perpendicular.

One horizontal fold gives 3 parallel lines (two edges and the crease), two folds give 5, three folds give 9. The gaps double each time.

Notation: arrow marks (>) show parallel lines (>> for a second set), and a small square marks a right angle.

5.5 Transversals

A line t that crosses two lines l and m is a transversal. It forms 8 angles, but because of the four pairs of vertically opposite angles there are at most 4 different measures.

5.6 Corresponding Angles

Angles in the same position at the two crossings (∠1 and ∠5, ∠2 and ∠6, ∠3 and ∠7, ∠4 and ∠8) are corresponding angles.

If the corresponding angles are equal, the lines are parallel. And if the lines are parallel, the corresponding angles are equal.

Activity 3: draw line l, a transversal t making 60° with it, then a line m also making 60° with t. The lines l and m come out parallel.

5.7 Drawing Parallel Lines

Ruler and set square: slide the set square along a ruler. Every line drawn makes the same angle (90°) with the ruler's line, so the corresponding angles are equal and the lines are parallel.

Paper folding: fold a crease t perpendicular to l through point A, then fold a crease m perpendicular to t through A. Now l and m both make 90° with t, so they are parallel.

5.8 Alternate Angles and Interior Angles

In Fig. 5.25, ∠d is the alternate angle of ∠f. To reach it: take the corresponding angle of ∠f (that is ∠b), then its vertically opposite angle (∠d).

Alternate angles formed by a transversal on parallel lines are equal.

Interior angles on the same side of the transversal add up to 180°.

5.9 Parallel Illusions

Some drawings make parallel lines look slanted or curved. Our eyes can be fooled, which is why we use angles to check whether lines are parallel.

Worked Examples

All eight angles from one (Example 1). Parallel lines with ∠6 = 135°. Its corresponding and vertically opposite angles are all 135° (∠2, ∠4, ∠6, ∠8). ∠5 forms a linear pair with ∠6, so ∠5 = 180° – 135° = 45°, and likewise ∠1, ∠3, ∠7 = 45°.

Are the lines parallel? (Example 2). ∠a = 120°, so its linear pair ∠b = 60°. The corresponding angle ∠f is 70°. Since 60° ≠ 70°, the lines are not parallel.

Interior angles (Example 3). ∠3 = 50°, so its linear pair ∠2 = 130°, and the corresponding angle ∠6 = 130°. Notice ∠3 + ∠6 = 180°.

Parallelogram ABCD (Example 4). AB ∥ CD and AD ∥ BC, ∠DAC = 65° and ∠ADC = 60°. Interior angles: ∠DAB = 180° – 60° = 120°, so ∠CAB = 120° – 65° = 55°. Also ∠BCD = 180° – 60° = 120° and ∠ABC = 60°.

A perpendicular pair. If one angle at an intersection is 90°, its linear pair is 180° – 90° = 90° and the vertically opposite angles are 90° too, so all four are right angles.

Figure it Out Hints

Start from the given angle, then use linear pair (sum 180°), vertically opposite (equal), corresponding (equal on parallel lines) or alternate (equal on parallel lines) step by step, writing the reason at each step.

For a bent path between parallel lines (Fig. 5.35), draw an extra line parallel to the given lines through the corner point, then split the angle.

To test whether lines are parallel, compare a pair of corresponding angles. Equal means parallel.

Maths Words

Plane (तल): a flat surface extending in all directions

Intersecting lines (प्रतिच्छेदी रेखाएँ): lines that meet at a point

Linear pair (रैखिक युग्म): adjacent angles adding to 180°

Vertically opposite angles (शीर्षाभिमुख कोण): opposite angles at an intersection; equal

Straight angle (ऋजु कोण): 180°

Right angle (समकोण): 90°

Perpendicular lines (लंब रेखाएँ): lines meeting at 90°

Parallel lines (समांतर रेखाएँ): lines on a plane that never meet

Transversal (तिर्यक रेखा): a line crossing two other lines

Corresponding angles (संगत कोण): angles in matching positions at the two crossings

Alternate angles (एकांतर कोण): angles on opposite sides of the transversal, between the lines

Interior angles (अंतःकोण): angles between the two lines on the same side of the transversal

Set square (सेट स्क्वायर): a triangular tool used to draw right angles and parallel lines

Proof (उपपत्ति/प्रमाण): a justification by reasoning

Common Mistakes

Assuming lines are parallel because they look parallel. Always check angles.

Using "corresponding angles are equal" when the lines are not given as parallel.

Mixing up a linear pair (sum 180°) with vertically opposite angles (equal).

Calling two lines on different surfaces (table and wall) parallel. Parallel lines must be on the same plane.

Adding interior angles on opposite sides of the transversal; the 180° rule is for the same side.

Maths in Daily Life

Railway tracks, window grills, ladders, zebra crossings, bookshelves and road markings all use parallel and perpendicular lines. Carpenters and architects use set squares and the "equal corresponding angles" idea to keep shelves and walls parallel.

💡 Key Learning Points

✓When two lines intersect, linear pairs add up to 180° and vertically opposite angles are equal, proved by reasoning
✓Perpendicular lines meet at 90°; parallel lines lie on the same plane and never meet
✓A transversal crossing two lines forms 8 angles with at most 4 different measures
✓Lines are parallel exactly when corresponding angles are equal; on parallel lines, alternate angles are equal and same-side interior angles add to 180°
✓Parallel lines can be drawn with a ruler and set square or by folding paper perpendicular twice

👨‍🏫 Teaching Tips

→Begin with the paper-folding activity so students discover intersecting, parallel and perpendicular creases themselves
→Have pairs draw four sets of intersecting lines and measure all angles, then discuss why the sums are not always exactly 180°
→Use tracing paper to slide ∠a onto ∠b along a transversal, making the corresponding-angle idea concrete
→Demonstrate the ruler-and-set-square method and let every student draw three parallel lines
→Ask students to justify each angle step with a reason (linear pair, vertically opposite, corresponding, alternate) like a mini proof
→Use the parallel illusions at the end to discuss why we trust angle reasoning more than our eyes
→Collect classroom examples of parallel and perpendicular lines for a wall display

📋 Assessment Questions

Can the student find all four angles at an intersection when one angle is given?
Can the student identify corresponding, alternate and interior angles in a transversal figure?
Can the student decide, with a reason, whether two lines cut by a transversal are parallel?
Can the student find all eight angles when parallel lines are cut by a transversal and one angle is known?
Can the student describe how to draw a line through a point parallel to a given line?

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