Class 7 : Maths – Lesson 5. Parallel and Intersecting Lines

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On December 11, 2025

EXPLANATION AND ANALYSIS

🔵 Introduction: Lines We See Around Us

🧠 In our surroundings, we often notice straight paths, railway tracks, electric wires, road crossings, window grills, and notebook lines. Many of these are examples of lines that either run side by side or cross each other. Mathematics helps us describe and understand such lines clearly.

🌿 Examples from daily life
🔵 Railway tracks running side by side
🟢 Zebra crossings on roads
🟡 Crossing roads at a junction
🔴 Grills of windows and doors

This chapter introduces two important ideas related to lines: parallel lines and intersecting lines.

🟢 Meaning of a Line

🧠 A line is a straight path that extends endlessly in both directions.

🔹 It has no fixed length
🔹 It has no thickness

📌 Example
The edge of a ruler represents a part of a line.

💡 Concept:
Lines are basic geometric figures used to form shapes and angles.

🔵 Parallel Lines

🧠 Parallel lines are lines that lie in the same plane and never meet, no matter how far they are extended.

🔹 The distance between parallel lines remains the same
🔹 They do not cross each other

📌 Example
The two long sides of a railway track are parallel lines.

✏️ Note:
Parallel lines are always drawn straight and in the same direction.

🟢 Identifying Parallel Lines

🧠 Parallel lines can be identified by observing their direction and spacing.

🔹 Lines moving in the same direction
🔹 Equal distance between them throughout

📌 Examples from daily life
🔹 Lines in ruled notebooks
🔹 Opposite sides of a rectangle
🔹 Steps of a staircase

💡 Concept:
If two lines never intersect and maintain constant distance, they are parallel.

🔵 Intersecting Lines

🧠 Intersecting lines are lines that meet or cross each other at a point.

🔹 The point where they meet is called the point of intersection
🔹 Intersecting lines form angles

📌 Example
Two roads crossing at a junction are intersecting lines.

✏️ Note:
Intersecting lines always meet at exactly one point.

🟢 Angles Formed by Intersecting Lines

🧠 When two lines intersect, they form four angles.

🔹 Some angles may be equal
🔹 Angles help us understand the direction and shape of figures

📌 Example
At a road crossing, the angles formed between roads show how they meet.

💡 Concept:
The study of angles begins with intersecting lines.

🔵 Perpendicular Lines

🧠 A special case of intersecting lines is perpendicular lines.

🔹 They intersect at a right angle
🔹 The angle formed is 90 degrees

📌 Example
The vertical and horizontal edges of a book meet at right angles.

✏️ Note:
All perpendicular lines are intersecting lines, but not all intersecting lines are perpendicular.

🟢 Difference Between Parallel and Intersecting Lines

🧠 Understanding the difference helps in identifying them easily.

🔹 Parallel lines never meet
🔹 Intersecting lines always meet at one point

📌 Example
Railway tracks are parallel
Crossroads are intersecting

💡 Concept:
Meeting or not meeting is the key difference.

🟡 Lines in Geometrical Shapes

🧠 Many geometrical shapes are formed using parallel and intersecting lines.

🔹 A rectangle has pairs of parallel sides
🔹 A triangle has sides that intersect
🔹 A square has parallel as well as perpendicular sides

📌 Example
In a rectangle, opposite sides are parallel and adjacent sides intersect.

🔴 Common Mistakes to Avoid

🔴 Thinking that slightly slanted lines are parallel
🔴 Confusing intersecting lines with touching lines
🔴 Forgetting that perpendicular lines are also intersecting

✏️ Note:
Always imagine lines extending endlessly while identifying them.

🟢 Importance of Parallel and Intersecting Lines

🧠 Learning about these lines helps us:

🔹 Understand basic geometry
🔹 Identify shapes correctly
🔹 Learn angles easily
🔹 Prepare for higher geometry topics

These ideas are used widely in drawing, architecture, road design, and engineering.

📘 Summary

🔵 A line extends endlessly in both directions
🟢 Parallel lines never meet and remain equidistant
🟡 Intersecting lines meet at one point
🔴 Perpendicular lines intersect at right angles
🔵 Intersecting lines form angles
🟢 Many shapes are formed using these lines

📝 Quick Recap

📝 Quick Recap
🔵 Parallel lines run side by side and never meet
🟢 Intersecting lines cross at one point
🟡 Perpendicular lines form right angles
🔴 Lines help form shapes and angles
🔵 These concepts are used in daily life

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TEXTBOOK QUESTIONS

🔵 ACROSS THE LINE

🔒 ❓ Question
List all the linear pairs and vertically opposite angles you observe in Fig. 5.3.

📌 ✅ Answer:

🔹 Linear Pairs
🔸 ∠a and ∠b
🔸 ∠b and ∠c
🔸 ∠c and ∠d
🔸 ∠d and ∠a

🔹 Pairs of Vertically Opposite Angles
🔸 ∠a and ∠c
🔸 ∠b and ∠d

🔵 PARALLEL AND PERPENDICULAR LINE

🔒 ❓ Question 1
Draw some lines perpendicular to the lines given on the dot paper in Fig. 5.10.

📌 ✅ Answer:

🔹 A line perpendicular to a given line makes a right angle (90°) with it.

🔹 For each given line in Fig. 5.10:
🔸 If the line is horizontal, draw a vertical line through any dot on it.
🔸 If the line is vertical, draw a horizontal line through any dot on it.
🔸 If the line is slanting, draw another slanting line such that the angle between them is 90°, using the dot grid for accuracy.

🔹 More than one correct perpendicular line is possible.

🔒 ❓ Question 2(a)
How did you spot the perpendicular lines?

📌 ✅ Answer:

🔹 Perpendicular lines were identified by checking the angle between the lines.
🔸 If the angle formed is 90°, the lines are perpendicular.

🔹 On the grid, one line is vertical and the other is horizontal, forming a right angle.

🔹 The right angle is marked using a square symbol at the point of intersection.

🔒 ❓ Question 2(b)
How did you spot the parallel lines?

📌 ✅ Answer:

🔹 Parallel lines were identified because they never meet, even when extended.

🔹 On the grid, parallel lines:
🔸 Have the same direction or slope.
🔸 Maintain a constant distance between them.

🔹 Lines with the same arrow notation (single arrow, double arrow, etc.) were marked as parallel.

🔒 ❓ Question 3
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.

📌 ✅ Answer:

🔹 Choose two or more line segments moving in the same direction on the dot paper.

🔹 Ensure that:
🔸 Each segment has dots as endpoints.
🔸 The segments do not intersect.
🔸 The distance between them remains the same.

🔹 Possible sets include:
🔸 Horizontal parallel line segments of different lengths.
🔸 Vertical parallel line segments.
🔸 Slanting parallel line segments with the same slope.

🔒 ❓ Question 4
Using your sense of how parallel lines look, try to draw lines parallel to the line segments on this dot paper.

🔒 ❓ Question 4(a)
Did you find it challenging to draw some of them?

📌 ✅ Answer:

🔹 Yes, some of the line segments were challenging to draw parallel to.
🔸 This difficulty was mainly due to their slanting direction on the dot paper.

🔒 ❓ Question 4(b)
Which ones?

📌 ✅ Answer:

🔹 The most challenging lines were the slanted line segments, such as:
🔸 Line b
🔸 Line c
🔸 Line d
🔸 Line h

🔹 These lines do not follow a simple horizontal or vertical direction, so judging their exact slope is harder.

🔒 ❓ Question 4(c)
How did you do it?

📌 ✅ Answer:

🔹 I observed the direction and slope of the given line carefully.

🔹 Then I counted the number of dots moved horizontally and vertically between the endpoints.

🔹 Using the same movement pattern, I drew another line with dots as endpoints so that:
🔸 Both lines had the same direction.
🔸 The distance between them remained constant.

🔹 This ensured that the new line was parallel to the given line.

🔒 ❓ Question 5
In Fig. 5.13, which line is parallel to line a — line b or line c? How do you decide this?

📌 ✅ Answer:

🔹 Line b is parallel to line a.

🔹 This is decided by observing that:
🔸 Line a and line b have the same direction (slope).
🔸 They would never meet, even if extended further.

🔹 Line c has a different direction, so it is not parallel to line a.

🔵 DRAWING PARALLEL LINES

🔒 ❓ Question
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.

📌 ✅ Answer:

🔹 Yes, a line parallel to l can be drawn through point A using the tools in a geometry box.

🔹 Method using ruler and set-square:

🔸 Place the ruler along the given line l.

🔸 Hold the ruler firmly so that it does not move.

🔸 Place a set-square against the ruler.

🔸 Slide the set-square upward carefully until one edge of the set-square passes through point A.

🔸 Draw a straight line along that edge of the set-square through point A.

🔹 The drawn line passes through A and has the same direction as line l, so it is parallel to l.

🔵 ALTERNATE ANGLES

🔒 ❓ Question 1
Find the angles marked below.

📌 ✅ Answer:

🔹 (a)
🔸 Given angle = 48°
🔸 a and 48° form a linear pair
🔸 a + 48° = 180°
🔸 a = 180° − 48°
🔸 a = 132°

🔹 (b)
🔸 Given angle = 52°
🔸 b is corresponding to 52°
🔸 b = 52°

🔹 (c)
🔸 99° and 81° form a linear pair on the top line
🔸 c corresponds to the angle 99°
🔸 c = 99°

🔹 (d)
🔸 81° corresponds to angle d
🔸 d = 81°

🔹 (e)
🔸 Given angle = 69°
🔸 e and 69° form a linear pair
🔸 e + 69° = 180°
🔸 e = 180° − 69°
🔸 e = 111°

🔹 (f)
🔸 Given angle = 132°
🔸 f is corresponding to 132°
🔸 f = 132°

🔹 (g)
🔸 Given angle = 58°
🔸 g is vertically opposite to 58°
🔸 g = 58°

🔹 (h)
🔸 Given angle = 120°
🔸 h is interior angle on the same side of the transversal
🔸 h + 120° = 180°
🔸 h = 180° − 120°
🔸 h = 60°

🔹 (i)
🔸 Given angle on straight line = 70°
🔸 Adjacent angle on the straight line = 180° − 70°
🔸 = 110°
🔸 i is corresponding to this angle
🔸 i = 110°

🔹 (j)
🔸 Given angle = 124°
🔸 j forms a linear pair with 124°
🔸 j + 124° = 180°
🔸 j = 180° − 124°
🔸 j = 56°

🔒 ❓ Question 2
Find the angle represented by a.

📌 ✅ Answer:

🔹 (i) Fig. 5.31 (top-left)
🔸 The given 42° is an angle formed by a transversal with a parallel line.
🔸 The adjacent angle on the straight line forms a linear pair with 42°.
🔸 a + 42° = 180°
🔸 a = 180° − 42°
🔸 a = 138°

🔹 (ii) Fig. 5.31 (top-right)
🔸 The given angle is 62°.
🔸 The adjacent angle on the straight line = 180° − 62° = 118°.
🔸 Angle a is corresponding to this adjacent angle.
🔸 a = 118°

🔹 (iii) Fig. 5.31 (bottom-left)
🔸 The angle between the transversal and the top parallel line is 110°.
🔸 The acute angle with the same transversal = 180° − 110° = 70°.
🔸 The angle between the two transversals is 35°.
🔸 a = 70° + 35°
🔸 a = 105°

🔹 (iv) Fig. 5.31 (bottom-right)
🔸 The vertical line is perpendicular to the base line.
🔸 The angle between the slanted line and the base line is 67°.
🔸 a + 67° = 90°
🔸 a = 90° − 67°
🔸 a = 23°

🔒 ❓ Question 3
In the figures below, what angles do x and y stand for?

📌 ✅ Answer:

🔹 Left figure (Fig. 5.32)
🔸 The vertical line is perpendicular to the horizontal line.
🔸 The angle between the slanted line and the vertical line is 65°.
🔸 The angle between the slanted line and the horizontal line = 90° − 65° = 25°.
🔸 x is this acute angle on the lower parallel line.
🔸 x = 25°
🔸 y is the corresponding acute angle on the upper parallel line.
🔸 y = 25°

🔹 Right figure (Fig. 5.32)
🔸 One transversal makes 53° with the lower parallel line.
🔸 The other transversal makes 78° with the lower parallel line.
🔸 The supplementary angle to 78° = 180° − 78° = 102°.
🔸 The angle between the two transversals at the top intersection
🔸 x = 102° − 53°
🔸 x = 49°

🔒 ❓ Question 4
In Fig. 5.33, ∠ABC = 45° and ∠IKJ = 78°. Find ∠GEH, ∠HEF, ∠FED.

📌 ✅ Answer:

🔹 The slanted line through B and E makes 45° with the top horizontal line.
🔸 It makes the same angle with the bottom horizontal line (corresponding angles).

🔹 ∠GEH
🔸 GE is along the bottom horizontal line.
🔸 EH is the slanted line.
🔸 ∠GEH = 45°

🔹 The line through K and E makes 78° with the horizontal line.

🔹 ∠FED
🔸 EF with ED forms the acute corresponding angle.
🔸 ∠FED = 78°

🔹 ∠HEF
🔸 Angle between the two slanted lines
🔸 = 78° − 45°
🔸 ∠HEF = 33°

🔒 ❓ Question 5
In Fig. 5.34, AB is parallel to CD and CD is parallel to EF. Also, EA is perpendicular to AB. If ∠BEF = 55°, find the values of x and y.

📌 ✅ Answer:

🔹 AB ∥ CD ∥ EF, and BE is a transversal cutting these parallel lines.

🔹 Given ∠BEF = 55° is the acute angle between BE and EF.

🔹 The angle on the other side at the same intersection (linear pair) is:
🔸 55° + 125° = 180°
🔸 125° = 180° − 55°

🔹 The angles marked x° (at B) and y° (at D) are in the same corresponding position as this obtuse angle.
🔸 So, x = 125°
🔸 So, y = 125°

📌 ✅ Final:
🔹 x = 125°
🔹 y = 125°

🔒 ❓ Question 6
What is the measure of angle ∠NOP in Fig. 5.35?
[Hint: Draw lines parallel to LM and PQ through points N and O.]

📌 ✅ Answer:

🔹 LM ∥ PQ (same arrow marks), so we can draw a vertical line through O parallel to them (as the hint says).

🔹 At M, the angle between LM and MN is 40°.
🔸 By corresponding angles, MN makes 40° with the vertical line through N as well.

🔹 At N, the angle between MN and NO is 96°.
🔸 MN is 40° to one side of the vertical, and NO is to the other side.
🔸 So, 40° + (angle between NO and vertical) = 96°
🔸 angle between NO and vertical = 96° − 40°
🔸 angle between NO and vertical = 56°

🔹 At P, the angle between PQ and PO is 52°.
🔸 So OP makes 52° with the vertical line through O (corresponding direction).

🔹 Now at O, ∠NOP is the angle between ON and OP.
🔸 ON is 56° to one side of the vertical.
🔸 OP is 52° to the other side of the vertical.
🔸 ∠NOP = 56° + 52°
🔸 ∠NOP = 108°

📌 ✅ Final:
🔹 ∠NOP = 108°

Class 7 : Maths – Lesson 5. Parallel and Intersecting Lines

EXPLANATION AND ANALYSIS

🔵 Introduction: Lines We See Around Us

🟢 Meaning of a Line

🔵 Parallel Lines

🟢 Identifying Parallel Lines

🔵 Intersecting Lines

🟢 Angles Formed by Intersecting Lines

🔵 Perpendicular Lines

🟢 Difference Between Parallel and Intersecting Lines

🟡 Lines in Geometrical Shapes

🔴 Common Mistakes to Avoid

🟢 Importance of Parallel and Intersecting Lines

📘 Summary

📝 Quick Recap

TEXTBOOK QUESTIONS

OTHER IMPORTANT QUESTIONS

🔵 Section A – Very Short Answer (1 × 6 = 6 marks)

🟢 Section B – Short Answer I (2 × 6 = 12 marks)

🟡 Section C – Short Answer II (3 × 10 = 30 marks)

🔴 Section D – Long Answer (4 × 8 = 32 marks)

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