✂ Intersecting ⊥ Perpendicular ∥ Parallel 📄 Paper Folding 📏 Transversal ↔ Corresponding ✏ Drawing ⇅ Alternate 📌 Summary 🧪 MCQ 📝 Q&A
Class 7 · Ganita Prakash · Chapter 5

Parallel &
Intersecting Lines

Intersecting · Perpendicular · Parallel · Transversals · Corresponding & Alternate Angles

∠ Vertically Opposite
∥ Parallel Lines
t Transversal
⊥ Perpendicular
✂5.1 Intersecting Lines

Take a piece of square paper and fold it in different ways. Draw lines on the creases using a pencil and scale. You will notice different lines on the paper. Take any pair of lines and observe — do they meet? If not within the paper, would they meet if extended?

Definition

When a pair of lines meet each other at a point on a plane surface, we say that the lines intersect each other. The point where they meet is called the point of intersection.

📐 Angles Formed at Intersection

When two lines intersect, they form four angles. These angles have special relationships:

l m a b c d ∠a = ∠c (Vertically Opposite) ∠b = ∠d (Vertically Opposite)
Fig. 5.2 — Two intersecting lines forming four angles

🔄 Vertically Opposite Angles

Opposite angles formed by two intersecting lines. Always equal.
∠a = ∠c    ∠b = ∠d

📏 Linear Pair

Adjacent angles formed at intersection. Always add up to 180°.
∠a + ∠b = 180°    ∠b + ∠c = 180°

🎛 Interactive Angle Calculator

Enter angle ∠a and all other angles are calculated automatically!

°
∠a
120°
∠b = 180−a
60°
∠c = ∠a
120°
∠d = ∠b
60°
📋 Activity 1 — Measure the Angles
  • Draw two lines on a plain sheet so that they intersect.
  • Measure the four angles with a protractor.
  • Draw four such pairs of intersecting lines and measure the angles.
  • What patterns do you observe? Are vertically opposite angles always equal?
Proof (Mathematical Reasoning)

Since straight angles measure 180°: ∠a + ∠b = ∠a + ∠d = 180° → so ∠b = ∠d always.
Similarly: ∠b + ∠a = ∠b + ∠c = 180° → so ∠a = ∠c always.
This reasoning without measurement is called a proof in mathematics.

📏 Measurements and Geometry: When you measure angles, they may not add up to exactly 180° due to (1) instrument errors from improper use of a protractor, and (2) the thickness of drawn lines — ideal geometric lines have NO thickness. Geometry works with ideal lines and uses reasoning, not measurement.
🔍 Figure it Out — Fig. 5.3
a b c d e f List all linear pairs and VOA pairs
Fig. 5.3 — Three lines through one point forming 6 angles
Linear pairs: (a,b), (b,c), (c,d), (d,e), (e,f), (f,a)  |  VOA: (a,d), (b,e), (c,f)
⊥5.2 Perpendicular Lines

Can you draw a pair of intersecting lines such that all four angles are equal? If all four angles at an intersection are equal, each must measure 180° ÷ 4 = 90°.

Definition

Perpendicular lines are a pair of lines which intersect each other at right angles (90°). All four angles formed are equal to 90°. We write l ⊥ m.

l m 90° 90° 90° 90° l ⊥ m (l is perpendicular to m)
Fig. 5.4 — Perpendicular lines: all four angles = 90°
∥5.3 Between Lines & Parallel Lines

In Fig. 5.5, observe and describe how line segments meet or cross each other using mathematical words: a point, an endpoint, the midpoint, meet, intersect and the degree measure of each angle.

Example: Line segments FG and FH meet at the endpoint F at an angle of 115.3°.

Some pairs of line segments, even when extended, do not seem to meet. These lead us to the idea of parallel lines.

Definition

Parallel lines are a pair of lines that lie on the same plane and do not meet however far we extend them in both directions.

l m l ∥ m — arrow marks (›) show they are parallel
Parallel lines l and m — they never meet however far extended

📍 Parallel Lines Examples

  • Opposite edges of a ruler
  • Railway tracks
  • Lines on a notebook
  • Opposite edges of a door

✏ Notations

  • › (single arrow) = one set of parallel lines
  • » (double arrow) = second set of parallel lines
  • □ (small square) = perpendicular lines
🎨 Parallel lines in Art: Parallel lines are often used in artwork and shading to create depth and texture. Artists use evenly spaced parallel lines (called hatching) to show shadows and 3D effects.
📄5.4 Parallel & Perpendicular Lines in Paper Folding
📋 Activity 2 — Square Sheet of Paper
  • Opposite edges of the sheet → Parallel to each other.
  • Adjacent edges of the sheet → Perpendicular to each other. They meet and form right angles.
  • Fold the sheet horizontally in half → a new line is formed.
  • The new line is also parallel to the top and bottom edges (perpendicular to the vertical sides).
  • Make one more horizontal fold → more parallel lines. Each new fold creates another parallel line. Pattern: 1 fold → 3 lines, 2 folds → 5 lines, 3 folds → 7 lines...
  • Make a vertical fold → the new vertical line is perpendicular to all the horizontal lines.
  • Fold along a diagonal → try to find a fold that creates a line parallel to the diagonal line.
📋 Activity 3 — Triangle Folds (Fig. 5.8)
  • Take a square sheet. Fold in the middle and unfold.
  • Fold edges towards the centre line and unfold.
  • Fold the top-right and bottom-left corners onto the creased line to create triangles.
  • The triangles should not cross the crease lines.
  • Question: Are lines a, b and c parallel to p, q and r respectively? Why or why not?
a b p q a ∥ p, b ∥ q — corresponding fold lines are parallel
Fig. 5.8 — Paper folding: fold lines are parallel to corresponding lines on the other half
📏5.5 Transversals
Definition

When a line intersects two or more lines, it is called a transversal. In Fig. 5.14, line t is the transversal intersecting lines l and m.

l m t 1 2 3 4 5 6 7 8 8 angles formed | Max 4 distinct measures | 4 VOA pairs
Fig. 5.14 — Transversal t intersects lines l and m, forming 8 angles

🔢 8 Angles — 4 Distinct Measures

Since ∠1=∠3, ∠2=∠4 (VOA at l) and ∠5=∠7, ∠6=∠8 (VOA at m), there are maximum 4 distinct angle values.

🔄 4 Pairs of VOA

∠1 & ∠3  |  ∠2 & ∠4 (at line l)
∠5 & ∠7  |  ∠6 & ∠8 (at line m)

📊 Angle Relationships at a Glance

Pair TypeAnglesRule
VOA at l∠1 & ∠3 | ∠2 & ∠4Equal
VOA at m∠5 & ∠7 | ∠6 & ∠8Equal
Corresponding∠1&∠5 | ∠2&∠6 | ∠3&∠7 | ∠4&∠8Equal (if l ∥ m)
Alternate∠1&∠7 | ∠2&∠8 | ∠3&∠5 | ∠4&∠6Equal (if l ∥ m)
Interior same side∠3&∠5 | ∠4&∠6Sum = 180° (if l ∥ m)
↔5.6 Corresponding Angles

The transversal t forms two sets of angles — one with line l and another with line m. Angles in matching positions are called corresponding angles.

Pair 1

∠1 & ∠5

Both above their lines, right of t

Pair 2

∠2 & ∠6

Both above their lines, left of t

Pair 3

∠3 & ∠7

Both below their lines, left of t

Pair 4

∠4 & ∠8

Both below their lines, right of t

📋 Activity 3 — Trace the Angle
  • Draw a line l and a transversal t intersecting at X. Measure ∠a = 60°.
  • The linear pair gives 120° → already 2 distinct angles.
  • Mark a point Y on a second line m. Draw m through Y at 60° to t (same as ∠a).
  • ∠b = 60° = ∠a → Lines l and m appear parallel!
Key Theorem (Both Directions)

▶ If a transversal makes equal corresponding angles with a pair of lines → the lines are parallel.

▶ If a transversal intersects parallel lines → the corresponding angles are equal.

▶ If lines are NOT parallel → corresponding angles can never be equal.

📋 Activity 4 — Verify with Tracing Paper
  • In Fig. 5.19, parallel lines l and m are cut by transversal t.
  • Trace ∠a on tracing paper. Place it over ∠b (corresponding angle).
  • The angles align exactly — confirming corresponding angles are equal.
  • Check all four corresponding pairs with a protractor.
📋 Activity 5 — The Hard Challenge

In Fig. 5.20, lines l and m are NOT parallel. Try to draw a transversal that makes equal corresponding angles. You will find it is impossible! This confirms that non-parallel lines can never have equal corresponding angles.

✏5.7 Drawing Parallel Lines
📐 Method 1: Ruler + Set Square (Fig. 5.21)
  1. Draw a line l with a ruler.
  2. Place a set square on l and draw a line perpendicular to l.
  3. Slide the set square along l (without rotating) and draw another line perpendicular to l.
  4. Both new lines make 90° with l → their corresponding angles are equal → they are parallel to each other.
  5. You can also draw parallel lines using the long side of the set square (Fig. 5.22).
📄 Method 2: Paper Folding (Fig. 5.24)
Given line l (a crease), draw a parallel through point A
  • Step 1: Fold a line perpendicular to l passing through A → call it crease t.
  • Step 2: Fold a line perpendicular to t passing through A → call it line m.
  • Result: l ∥ m
  • Why? Both l and m are perpendicular to t → the transversal t makes 90° with both → corresponding angles are equal → l ∥ m.
💡 Key Insight: Two lines that are both perpendicular to the same line are always parallel to each other. This is because their corresponding angles (both 90°) with the transversal are equal.
✏ Figure it Out

In Fig. 5.23, draw a line parallel to line l passing through point A. How would you do it with tools from your geometry box?

Hint: Use a set square and ruler. Place the set square so one edge lies on l, draw a perpendicular through A (line t), then rotate the set square to draw a perpendicular to t through A. That gives the parallel line.
⇅5.8 Alternate Angles

In Fig. 5.25, when transversal t crosses lines l and m, some angles are on opposite sides of the transversal and between the lines. These are called alternate angles.

l m t a b c d e f g h ∠d and ∠f are alternate angles (highlighted) — both interior, opposite sides of t
Fig. 5.25 — Alternate angles ∠d & ∠f; also ∠c & ∠e are alternate pairs
How to Find Alternate Angle of ∠f

Step 1: Find corresponding angle of ∠f → that is ∠b
Step 2: Find vertically opposite angle of ∠b → that is ∠d
So ∠d is the alternate angle of ∠f
Since ∠f = ∠b (corresponding) and ∠b = ∠d (VOA) → ∠f = ∠d always

Key Theorem — Alternate Angles

Alternate angles formed by a transversal intersecting a pair of parallel lines are always equal to each other.

📐 Interior Angles on the Same Side

The angles between the parallel lines on the same side of the transversal (co-interior angles) always add up to 180°.
∠3 + ∠5 = 180°    ∠4 + ∠6 = 180°

📝 Worked Examples
Example 1 — Find all angles when ∠6 = 135°

Given: l ∥ m, transversal t, ∠6 = 135°

  • ∠2 = 135° (corresponding angle of ∠6, as l ∥ m)
  • ∠8 = 135° (vertically opposite to ∠6)
  • ∠4 = 135° (corresponding angle of ∠8)
  • ∴ ∠2, ∠4, ∠6, ∠8 = 135°
  • ∠5 = 180° − 135° = 45° (linear pair with ∠6)
  • ∴ ∠1, ∠3, ∠5, ∠7 = 45°
Example 2 — Are l and m parallel? (∠a = 120°, ∠f = 70°)

∠a = 120° → ∠b = 180° − 120° = 60° (linear pair)

∠b is the corresponding angle of ∠f. For l ∥ m, we need ∠b = ∠f.

But ∠b = 60° ≠ ∠f = 70° → Lines l and m are NOT parallel.

Example 3 — Find ∠6 when ∠3 = 50° (l ∥ m)

∠3 = 50° → ∠2 = 180° − 50° = 130° (linear pair)

∠2 and ∠6 are corresponding angles (l ∥ m) → ∠6 = 130°

∠3 + ∠6 = 180° → they are interior angles on the same side of the transversal.

Example 4 — AB ∥ CD, AD ∥ BC, ∠DAC = 65°, ∠ADC = 60°. Find ∠CAB, ∠ABC, ∠BCD
  • AB ∥ CD, transversal AD: interior angles → ∠ADC + ∠DAB = 180° → ∠DAB = 120°
  • ∠DAB = ∠DAC + ∠CAB → 120° = 65° + ∠CAB → ∠CAB = 55°
  • AD ∥ BC, transversal CD: ∠ADC + ∠BCD = 180° → ∠BCD = 120°
  • Similarly: ∠ABC = 60°
🔍 Figure it Out — Find Marked Angles

a° (with 48°)

a = 48° (corresponding angles, parallel lines)

b° (with 52°)

b = 52° (alternate angles)

d° (81°, 99°)

d = 99° (corresponding to 99°)

g° (58°, 122° ×2)

g = 58° (alternate angles)

h° (75°, 120°)

h = 180° − 120° = 60° (co-interior angles)

e° (97°, 83°, 69°)

e = 180° − 83° = 97° (co-interior)

👁5.9 Parallel Illusions

Some figures make lines appear non-parallel even when they actually are. These are called parallel illusions. The background pattern tricks our eyes.

Café Wall Illusion — the horizontal grey lines ARE parallel!
The two vertical lines ARE straight & parallel!
Lines appear bowed but are actually straight and parallel
🧠 Why do illusions work? Our brain processes the angles and context of surrounding lines to judge parallelism. When background lines radiate or create strong angles, the brain is "tricked" into seeing bowing or convergence that is not there. Use a ruler to check!
📌In a Nutshell

📌 Chapter 5 Summary

When two lines intersect, they form four angles. The vertically opposite angles are equal and the linear pairs add up to 180°.
When two lines intersect and all four angles = 90°, the lines are said to be perpendicular to each other (l ⊥ m).
When two lines never intersect on a plane, they are called parallel lines (l ∥ m). Notation: single arrow ›, double arrow ».
When a line t intersects another pair of lines, it is called a transversal. It forms 8 angles with a maximum of 4 distinct angle measures.
When a transversal intersects parallel lines → corresponding angles are equal. When corresponding angles are equal → lines are parallel. (Necessary AND sufficient!)
When a transversal intersects parallel lines → alternate angles are equal. (Find: corresponding angle → then vertically opposite angle.)
The interior angles on the same side formed by a transversal intersecting parallel lines always add up to 180°.

🧪 Multiple Choice Questions

1. Two lines intersect and one angle formed is 65°. What is the vertically opposite angle?
2. Two lines intersect. One angle is 110°. What is its linear pair?
3. Perpendicular lines intersect at what angle?
4. How many angles are formed when a transversal crosses two lines?
5. Which symbol is used to denote parallel lines in a figure?
6. A transversal crosses two parallel lines. One corresponding angle is 75°. What is the other?
7. Interior angles on the same side of a transversal (with parallel lines) always add up to:
8. l ∥ m with transversal t. ∠1 = 55°. What is the alternate angle?
9. Which of the following is NOT a property of parallel lines cut by a transversal?
10. A justification through reasoning in mathematics (without measurement) is called a:

📝 Questions & Answers

✏ 1 Mark
Q1. Define intersecting lines. 1M
Answer: When a pair of lines meet each other at a point on a plane surface, they are called intersecting lines. The point where they meet is called the point of intersection.
Q2. What are vertically opposite angles? 1M
Answer: Opposite angles formed by two intersecting lines are called vertically opposite angles. They are always equal.
Q3. What is a linear pair? 1M
Answer: Adjacent angles formed by two intersecting lines are called a linear pair. They always add up to 180°.
Q4. Define parallel lines. 1M
Answer: Parallel lines are a pair of lines that lie on the same plane and do not meet however far we extend them in both directions. Notation: l ∥ m.
Q5. Define a transversal. 1M
Answer: A transversal is a line that intersects two or more lines at distinct points.
Q6. What are perpendicular lines? 1M
Answer: Perpendicular lines are a pair of intersecting lines that form a right angle (90°) at their point of intersection. Written as l ⊥ m.
Q7. How many angles are formed when a transversal cuts two lines? 1M
Answer: 8 angles are formed. Since vertically opposite angles are equal, there are at most 4 distinct angle measures.
Q8. What is the maximum number of distinct angle measures formed when a transversal cuts two lines? 1M
Answer: Four distinct angle measures at most. (8 angles with 4 VOA pairs → 4 distinct values.)
📘 2 Mark
Q9. Two lines intersect and ∠a = 70°. Find ∠b, ∠c and ∠d. 2M
Answer:
∠b = 180° − 70° = 110° (linear pair with ∠a)
∠c = 70° (vertically opposite to ∠a) = 70°
∠d = 110° (vertically opposite to ∠b) = 110°
Q10. What are corresponding angles? Name all four corresponding pairs for transversal t on lines l and m. 2M
Answer: Angles in the same position at each intersection point are called corresponding angles.
Four pairs: ∠1 & ∠5, ∠2 & ∠6, ∠3 & ∠7, ∠4 & ∠8
Q11. l ∥ m with transversal t. ∠3 = 50°. Find ∠6 and explain. 2M
Answer:
∠2 = 180° − 50° = 130° (linear pair with ∠3)
∠6 = ∠2 = 130° (corresponding angles, l ∥ m)
Also: ∠3 + ∠6 = 50° + 130° = 180° → they are interior angles on the same side.
Q12. How can you draw a line parallel to l through a point A using paper folding? 2M
Answer:
Step 1: Fold a line perpendicular to l passing through A → crease t
Step 2: Fold a line perpendicular to t passing through A → line m
l ∥ m because both are perpendicular to t, so the transversal t makes equal corresponding angles (90°) with both lines.
Q13. ∠a = 120°, ∠f = 70°. Are the two lines parallel? Justify. 2M
Answer:
∠a = 120° → ∠b = 180° − 120° = 60° (linear pair)
∠b and ∠f are corresponding angles. For parallel lines, ∠b must equal ∠f.
But 60° ≠ 70°, so the lines are NOT parallel.
📗 3 Mark
Q14. Explain with proof why vertically opposite angles are always equal. 3M
Answer (Proof):
Let two lines intersect forming angles ∠a, ∠b, ∠c, ∠d (∠a and ∠c opposite; ∠b and ∠d opposite).

Since ∠a and ∠b form a straight angle: ∠a + ∠b = 180° … (1)
Since ∠a and ∠d form a straight angle: ∠a + ∠d = 180° … (2)
From (1) and (2): ∠a + ∠b = ∠a + ∠d → ∠b = ∠d

Similarly, ∠b + ∠a = ∠b + ∠c → ∠a = ∠c
This reasoning without measurement is a proof.
Q15. Explain how alternate angles are always equal when lines are parallel. 3M
Answer:
Let transversal t cut parallel lines l and m. Consider angle ∠f at line m.

Step 1: ∠b is the corresponding angle of ∠f → ∠f = ∠b (corresponding angles, l ∥ m)
Step 2: ∠d is the vertically opposite angle of ∠b → ∠b = ∠d
Therefore: ∠f = ∠d — the alternate angles are always equal.

This holds for any value of ∠f because it relies only on corresponding angles (parallel lines) and vertically opposite angles — both always true.
Q16. In Fig. 5.29, AB ∥ CD and AD ∥ BC. ∠DAC = 65°, ∠ADC = 60°. Find ∠CAB, ∠ABC and ∠BCD. 3M
Answer:
AB ∥ CD, transversal AD: interior angles → ∠ADC + ∠DAB = 180°
60° + ∠DAB = 180° → ∠DAB = 120°
∠DAB = ∠DAC + ∠CAB → 120° = 65° + ∠CAB → ∠CAB = 55°

AD ∥ BC, transversal CD: ∠ADC + ∠BCD = 180°
60° + ∠BCD = 180° → ∠BCD = 120°

Similarly, AD ∥ BC, transversal AB: ∠DAB + ∠ABC = 180°
120° + ∠ABC = 180° → ∠ABC = 60°
📕 5 Mark
Q17. Write all the properties of angles formed when a transversal cuts two parallel lines. Give examples. 5M
Answer:
  1. Corresponding Angles are Equal: ∠1=∠5, ∠2=∠6, ∠3=∠7, ∠4=∠8. E.g., if ∠1 = 60°, then ∠5 = 60°.
  2. Alternate Angles are Equal: ∠3=∠5, ∠4=∠6 (alternate interior); ∠1=∠7, ∠2=∠8 (alternate exterior). E.g., if ∠3 = 50°, then ∠5 = 50°.
  3. Co-interior (Same-side Interior) Angles are Supplementary: ∠3+∠6 = 180°, ∠4+∠5 = 180°. E.g., if ∠3 = 50°, then ∠6 = 130°.
  4. Vertically Opposite Angles at each intersection are Equal: ∠1=∠3, ∠2=∠4 (at l), ∠5=∠7, ∠6=∠8 (at m).
  5. Linear pairs at each intersection add up to 180°.
Converse: If any one of (1), (2), or (3) holds, then the lines are parallel.
Q18. l ∥ m cut by transversal t. ∠6 = 135°. Find all 8 angles and state the property used for each. 5M
Answer:
AngleValueProperty Used
∠6135°Given
∠8135°Vertically opposite to ∠6
∠2135°Corresponding to ∠6 (l ∥ m)
∠4135°Corresponding to ∠8 (l ∥ m)
∠545°Linear pair with ∠6 (180°−135°)
∠745°Vertically opposite to ∠5
∠145°Corresponding to ∠5 (l ∥ m)
∠345°Vertically opposite to ∠1