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Parallel and Intersecting Lines

NCERT Class 7 · Mathematics Based on NCERT Class 7 Mathematics textbook · Free CBSE study kit

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Chapter Notes

Overview
This chapter explores the relationships between lines on a plane surface, focusing on what happens when lines intersect, when they are perpendicular, and when they are parallel. It introduces key geometric concepts like vertically opposite angles, linear pairs, corresponding angles, alternate angles, and interior angles formed by a transversal. The one big idea a student must walk away with is that when a transversal intersects parallel lines, specific angle relationships (corresponding angles equal, alternate angles equal, interior angles on the same side summing to 180°) always hold true, and these relationships can be used to determine if lines are parallel.
Key Concepts (12)
Intersecting Lines
When a pair of lines meet each other at a point on a plane surface, we say that the lines intersect each other.
e.g. The two edges of a table top meeting at a corner are intersecting lines.
Linear Pair
Adjacent angles formed by two lines intersecting each other are called linear pairs. Linear pairs always add up to 180°.
e.g. In a crossroad, the angle between the road going north and the road going east is a linear pair with the angle between the road going north and the road going west; together they form a straight line of 180°.
Vertically Opposite Angles
Opposite angles formed by two lines intersecting each other are called vertically opposite angles. Vertically opposite angles are always equal to each other.
e.g. When two roads cross, the angles directly opposite each other (like the north-east angle and the south-west angle) are equal.
Perpendicular Lines
Perpendicular lines are a pair of lines which intersect each other at right angles (90°).
e.g. The adjacent edges of a square sheet of paper are perpendicular to each other.
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.
e.g. The opposite edges of a table top are parallel to each other.
Transversal
A line that intersects two different lines is called a transversal.
e.g. A railway track crossing two parallel roads is a transversal.
Corresponding Angles
Angles in the first set that correspond to angles in the second set based on their position when a transversal intersects two lines. For example, ∠1 and ∠5, ∠2 and ∠6, ∠3 and ∠7, ∠4 and ∠8 are corresponding angles.
e.g. In a ladder, the angles made by a rung (transversal) with the two side rails (lines) at the same relative position are corresponding angles.
Alternate Angles
Angles formed on opposite sides of the transversal and inside the two lines. For example, ∠d is the alternate angle of ∠f, and ∠c is the alternate angle of ∠e.
e.g. In a 'Z' shape formed by a transversal crossing two parallel lines, the angles inside the 'Z' are alternate angles.
Interior Angles
Angles formed between the two lines when a transversal intersects them. For example, ∠3 and ∠6 are interior angles on the same side of the transversal.
e.g. The angles inside a railway track between the two rails and a sleeper (transversal) are interior angles.
Proof
A justification in mathematics that shows a statement is always true through reasoning, not just measurement.
e.g. We proved that vertically opposite angles are always equal by reasoning that ∠a + ∠b = 180° and ∠a + ∠d = 180°, so ∠b = ∠d.
Plane Surface
A flat surface like a table top, a piece of paper, a blackboard, or a bulletin board.
e.g. The surface of a notebook is a plane surface.
Straight Angle
An angle that measures 180°, formed by a straight line.
e.g. The angle formed by a straight line drawn on paper is 180°.
Formulas & Equations (5)
∠a + ∠b = 180° (Linear pair)
Reads as: Angle a plus angle b equals 180 degrees
Variables: ∠a and ∠b are adjacent angles formed by intersecting lines
Use when: When two lines intersect and you need to find the measure of an adjacent angle given one angle
⚠ Students forget that linear pairs are adjacent angles and add up to 180°, not 90°.
∠a = ∠c (Vertically opposite angles)
Reads as: Angle a equals angle c
Variables: ∠a and ∠c are opposite angles formed by intersecting lines
Use when: When two lines intersect and you need to find the measure of the opposite angle
⚠ Students confuse vertically opposite angles with adjacent angles and add them instead of equating them.
Corresponding angles are equal when lines are parallel
Reads as: Corresponding angles are equal when lines are parallel
Variables: Corresponding angles are pairs like ∠1 and ∠5, ∠2 and ∠6, etc.
Use when: When a transversal intersects parallel lines and you need to find unknown angles
⚠ Students assume corresponding angles are equal even when lines are not parallel.
Alternate angles are equal when lines are parallel
Reads as: Alternate angles are equal when lines are parallel
Variables: Alternate angles are pairs like ∠d and ∠f, ∠c and ∠e
Use when: When a transversal intersects parallel lines and you need to find alternate angles
⚠ Students confuse alternate angles with corresponding angles.
Interior angles on the same side of the transversal sum to 180° when lines are parallel
Reads as: Interior angles on the same side of the transversal sum to 180 degrees when lines are parallel
Variables: Interior angles on the same side are pairs like ∠3 and ∠6
Use when: When a transversal intersects parallel lines and you need to find interior angles on the same side
⚠ Students think interior angles on the same side are equal instead of summing to 180°.
How It Works
Finding unknown angles when two lines intersect
  1. Step 1: Identify the given angle measure.
  2. Step 2: Use the linear pair property: adjacent angles sum to 180°.
  3. Step 3: Use the vertically opposite angles property: opposite angles are equal.
  4. Step 4: Continue applying these properties until all angles are found.
💡 Always check if the angles are adjacent (linear pair) or opposite (vertically opposite) before applying the rule.
Determining if two lines are parallel using a transversal
  1. Step 1: Identify the transversal intersecting the two lines.
  2. Step 2: Measure or find the corresponding angles.
  3. Step 3: If the corresponding angles are equal, the lines are parallel.
  4. Step 4: Alternatively, check if alternate angles are equal or if interior angles on the same side sum to 180°.
💡 Corresponding angles must be equal for lines to be parallel; if they are not equal, the lines are not parallel.
Drawing parallel lines using a set square and ruler
  1. Step 1: Draw a line l with a scale.
  2. Step 2: Place a set square so that one side is along line l.
  3. Step 3: Hold a ruler against the other side of the set square so it doesn't move.
  4. Step 4: Slide the set square along the ruler to the desired position.
  5. Step 5: Draw a line along the edge of the set square. This line is parallel to line l.
💡 The set square ensures the new line makes a 90° angle with the ruler, so the corresponding angles with line l are equal (90°), confirming parallelism.
Drawing a line parallel to a given line through a point using paper folding
  1. Step 1: Fold the paper to create a crease perpendicular to the given line l, passing through point A. Call this crease t.
  2. Step 2: Fold the paper again to create a crease perpendicular to t, passing through point A. Call this crease m.
  3. Step 3: The crease m is parallel to line l.
💡 Lines l and m are both perpendicular to t, so they are parallel to each other because corresponding angles are 90°.
Finding alternate angles using corresponding and vertically opposite angles
  1. Step 1: Identify the given angle, say ∠f.
  2. Step 2: Find its corresponding angle, which is ∠b.
  3. Step 3: Find the vertically opposite angle of ∠b, which is ∠d.
  4. Step 4: ∠d is the alternate angle of ∠f, and they are equal.
💡 Alternate angles are found by first finding the corresponding angle and then its vertically opposite angle.
Exam Traps — Don't Make These Mistakes
⚠ Assuming all intersecting lines are perpendicular
✗ If two lines intersect, they are perpendicular.
→ Intersecting lines can intersect at any angle; perpendicular lines are a special case where all four angles are 90°.
✓ Two lines intersect at a point, but they are perpendicular only if they intersect at right angles (90°).
⚠ Confusing linear pairs with vertically opposite angles
✗ ∠a and ∠c are a linear pair because they are opposite each other.
→ Linear pairs are adjacent angles that sum to 180°, not opposite angles. Vertically opposite angles are equal, not supplementary.
✓ ∠a and ∠b are a linear pair because they are adjacent and sum to 180°. ∠a and ∠c are vertically opposite angles and are equal.
⚠ Assuming corresponding angles are equal even when lines are not parallel
✗ Since ∠a and ∠e are corresponding angles, they are equal.
→ Corresponding angles are equal only when the lines intersected by the transversal are parallel. If the lines are not parallel, corresponding angles are not equal.
✓ If lines l and m are parallel, then corresponding angles are equal. If they are not parallel, corresponding angles are not equal.
⚠ Mixing up alternate angles and interior angles on the same side
✗ Alternate angles on the same side of the transversal sum to 180°.
→ Alternate angles are on opposite sides of the transversal and are equal. Interior angles on the same side of the transversal sum to 180°.
✓ Alternate angles are equal. Interior angles on the same side of the transversal sum to 180°.
⚠ Thinking parallel lines must be of equal length
✗ Lines AB and CD are not parallel because AB is longer than CD.
→ Parallel lines are defined by never meeting, not by having equal length. Line segments can be of different lengths and still be parallel.
✓ Parallel lines are lines that never meet, regardless of their length. Line segments of different lengths can be parallel if they lie on parallel lines.
Compare & Contrast
Whether they meet
Intersecting Lines
Parallel Lines
Intersecting lines meet at a point; parallel lines never meet, no matter how far they are extended.
Relationship between angles
Linear Pair
Vertically Opposite Angles
Linear pairs are adjacent angles that sum to 180°; vertically opposite angles are opposite angles that are equal.
Position relative to the transversal and lines
Corresponding Angles
Alternate Angles
Corresponding angles are on the same side of the transversal and in the same relative position; alternate angles are on opposite sides of the transversal and inside the two lines.
Exam Focus
Marks pattern: 2 MCQs (1 mark each), 1 SA (3 marks), 1 LA (5 marks)
Most asked: Finding unknown angles using linear pair and vertically opposite angles properties · Identifying and using corresponding angles when lines are parallel · Identifying and using alternate angles when lines are parallel · Using interior angles on the same side to find unknown angles · Determining if lines are parallel based on angle relationships
Don't skip: A long answer question where a transversal intersects parallel lines and students must find all eight angles given one angle measure; A question involving multiple parallel lines and transversals requiring step-by-step angle calculation
One-Line Revision (read aloud before the exam)
  1. When two lines intersect, vertically opposite angles are always equal.
  2. Linear pairs are adjacent angles formed by intersecting lines and always sum to 180°.
  3. Perpendicular lines intersect at right angles (90°).
  4. Parallel lines lie on the same plane and never meet, no matter how far extended.
  5. A transversal is a line that intersects two or more lines.
  6. When a transversal intersects parallel lines, corresponding angles are equal.
  7. When a transversal intersects parallel lines, alternate angles are equal.
  8. When a transversal intersects parallel lines, interior angles on the same side sum to 180°.
  9. If corresponding angles formed by a transversal are equal, the lines are parallel.
  10. If alternate angles formed by a transversal are equal, the lines are parallel.
  11. If interior angles on the same side of a transversal sum to 180°, the lines are parallel.
  12. Vertically opposite angles are equal regardless of whether the lines are parallel or not.
  13. A straight angle measures 180°.
  14. Parallel lines are marked with arrow symbols (>, >>, etc.) and perpendicular lines with a square symbol.
  15. In geometry, we use reasoning (proof) rather than just measurement to establish relationships.

MCQs — 10 Questions with Answers

Q1. When two straight lines intersect, how many angles are formed at the point of intersection?

  • A. 2 angles
  • B. 3 angles
  • C. 4 angles ✓
  • D. 5 angles

Answer: C — Two intersecting lines create exactly 4 angles at their point of intersection, arranged around the intersection point.

Q2. If one angle formed by two intersecting lines is 75°, what is the measure of the vertically opposite angle?

  • A. 75° ✓
  • B. 105°
  • C. 90°
  • D. 180°

Answer: A — Vertically opposite angles are always equal, so if one angle is 75°, the opposite angle must also be 75°.

Q3. Two adjacent angles formed by intersecting lines measure 130° and 50°. What is their relationship?

  • A. They are vertically opposite angles
  • B. They form a linear pair ✓
  • C. They are perpendicular
  • D. They are parallel

Answer: B — Adjacent angles that sum to 180° (130° + 50° = 180°) form a linear pair, which is the defining property of angles on a straight line.

Q4. Which pair of lines intersects at right angles?

  • A. Parallel lines
  • B. Perpendicular lines ✓
  • C. Adjacent lines
  • D. Intersecting lines at 60°

Answer: B — Perpendicular lines are defined as lines that intersect each other at right angles (90°).

Q5. Identify which statement about parallel lines is correct.

  • A. Parallel lines lie on different planes and never meet
  • B. Parallel lines lie on the same plane and never meet, however far they are extended ✓
  • C. Parallel lines must intersect at one point
  • D. Parallel lines form right angles when extended

Answer: B — Parallel lines must be on the same plane and the key property is that they never meet no matter how far extended.

Q6. Ramesh draws two intersecting lines and measures the four angles. He gets 120°, 60°, 120°, and 60°. Which geometric property does this demonstrate?

  • A. Only linear pairs exist in intersecting lines
  • B. All angles formed must be equal
  • C. Vertically opposite angles are equal and linear pairs sum to 180° ✓
  • D. Perpendicular lines always form these specific angles

Answer: C — The pattern 120°, 60°, 120°, 60° shows opposite angles are equal (120° and 60° pairs) and adjacent angles sum to 180°, confirming both properties.

Q7. Which of the following is NOT a property of two lines intersecting on a plane?

  • A. They meet at exactly one point
  • B. Vertically opposite angles are equal
  • C. Adjacent angles sum to 180°
  • D. They can meet at two different points ✓

Answer: D — Two straight lines can intersect at exactly one point only; they cannot meet at two different points as this would contradict the definition of a straight line.

Q8. In a classroom, Priya observes that the top and bottom edges of the blackboard never meet, and the left and right edges also never meet. What can she conclude about these edges?

  • A. All edges are perpendicular to each other
  • B. Top and bottom edges are parallel; left and right edges are parallel ✓
  • C. The edges are on different planes so they cannot be parallel
  • D. The edges will eventually meet if extended beyond the blackboard

Answer: B — Opposite edges of a rectangular blackboard lying on the same plane that never meet are parallel lines, making two pairs of parallel edges.

Q9. When you measure angles formed by two intersecting lines using a protractor, sometimes linear pairs don't add up to exactly 180°. What is the main reason for this difference?

  • A. The lines are not actually straight
  • B. The geometric property is wrong
  • C. Measurement errors occur due to improper protractor use and line thickness ✓
  • D. Perpendicular lines cannot form linear pairs

Answer: C — Protractor measurement errors and the thickness of drawn lines (though ideal geometry lines have no thickness) cause slight variations from the theoretical 180°.

Q10. Two lines on a flat paper are perpendicular. A third line is drawn parallel to the first line. What is the relationship between the third line and the second line?

  • A. They are also perpendicular ✓
  • B. They are parallel
  • C. They must intersect at 60°
  • D. They cannot be determined without measurement

Answer: A — If line 1 ⊥ line 2, and line 3 ∥ line 1, then line 3 must also be perpendicular to line 2 because parallel lines have the same angle relationships with other lines.

Flashcards

What are vertically opposite angles?

Angles formed opposite each other when two lines intersect, and they are always equal to each other.

Define linear pair of angles.

Two adjacent angles formed by two intersecting lines that together form a straight angle and sum to 180°.

What is a perpendicular line?

A pair of lines that intersect each other at right angles (90°).

Define parallel lines.

Two lines on the same plane that never meet, no matter how far they are extended in either direction.

If two intersecting lines form a 45° angle, what is the adjacent angle?

The adjacent angle is 135°, because linear pairs always sum to 180°.

How many angles are formed when two lines intersect?

Exactly four angles are formed at the point of intersection.

Why do measurements sometimes not match geometric properties?

Due to measurement errors with protractors, the thickness of drawn lines, and the fact that ideal geometric lines have no thickness.

What does the arrow notation (>) mean in parallel lines?

The arrow mark is used to show that a set of lines is parallel to each other.

Can two straight lines intersect at more than one point?

No, two straight lines can intersect at exactly one point only.

Name two pairs of lines from your classroom that are parallel.

Any two correct examples: opposite edges of blackboard, top and bottom edges of a window frame, edges of a desk top, or parallel lines on the floor.

Important Board Questions

Define vertically opposite angles and linear pairs. Give one example of each from intersecting lines. [2 marks]

State the definition of vertically opposite angles (opposite angles when two lines intersect, always equal) and linear pairs (adjacent angles on a straight line, sum to 180°). Identify these in a simple diagram like Fig. 5.2 where ∠a and ∠c are vertically opposite, and ∠a and ∠b are a linear pair.

Two lines intersect such that one angle measures 110°. Find the measures of all four angles formed, and explain which angles are equal and why. [3 marks]

Use the linear pair property: if one angle is 110°, the adjacent angle is 180° - 110° = 70°. Then apply vertically opposite angles property: the angle opposite to 110° is also 110°, and the angle opposite to 70° is also 70°. Justify using: ∠a + ∠b = 180° (linear pair) and ∠a = ∠c (vertically opposite).

Explain why geometrical properties like 'linear pairs sum to 180°' are proven through reasoning rather than just measurement. In your answer, discuss the relationship between ideal geometry and real-world drawings, and give two reasons why measurements might not perfectly match the property. [5 marks]

Explain that ideal lines in geometry have no thickness and are infinitely straight, but real drawn lines have thickness and imperfections. Two key reasons: (1) measurement errors from improper protractor use, (2) thickness of drawn lines. Use example: a straight angle is always 180° by definition, not by measurement. Connect this to why geometry is reliable for architecture and engineering despite measurement variations.

Assertion-Reason Questions for Parallel and Intersecting Lines

Q1. Assertion (A):

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

Reason (R):

When two lines intersect, they form two pairs of adjacent angles, and each pair of adjacent angles is supplementary (sums to 180°).

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are true. The reason correctly explains why vertically opposite angles must be equal: since ∠a + ∠b = 180° and ∠c + ∠b = 180°, we get ∠a = ∠c. The reason is indeed the correct explanation for the assertion.

Q2. Assertion (A):

Perpendicular lines always intersect at right angles.

Reason (R):

Perpendicular lines form four angles that are all equal and each measures 90°.

Answer: B — Both A and R are true, but R is NOT the correct explanation of A — Both the assertion and reason are true statements. However, the reason is not the correct explanation for the assertion. The assertion defines perpendicular lines as intersecting at right angles (90°), while the reason describes a consequence of perpendicularity (all four angles being 90°). The reason doesn't explain why perpendicular lines intersect at right angles; it's rather a result of the definition.

Q3. Assertion (A):

Two lines lying on the same plane can be either parallel or intersecting.

Reason (R):

Parallel lines never meet, while intersecting lines meet at exactly one point on a plane.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are true. The reason correctly explains the assertion by distinguishing between the two mutually exclusive possibilities for two lines on the same plane: they either never meet (parallel) or meet at one point (intersecting). The reason is the correct explanation.

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Case-Based Questions (CBQ) for Parallel and Intersecting Lines

Case Study:

A design team is creating a floor plan for a new shopping mall. They need to draw corridors such that some corridors will meet at a central plaza (intersecting corridors), while other corridors are designed to never meet (parallel corridors). The team marks corridor AB and corridor CD as perpendicular to each other at the central plaza point P. They also mark corridor EF parallel to corridor AB. One assistant measures the angles and finds that corridor AB makes a 90° angle with corridor CD at point P.

Q1. What is the measure of all four angles formed where corridors AB and CD intersect at point P?

  • A. All angles are 90° ✓
  • B. Two angles are 90° and two are 180°
  • C. All angles are different
  • D. Two angles are 45° and two are 135°

Answer: A

Q2. Since corridor EF is parallel to corridor AB, which statement is true?

  • A. Corridor EF will intersect corridor CD at some point
  • B. Corridor EF will never intersect corridor AB
  • C. Corridor EF will be perpendicular to corridor CD ✓
  • D. Corridor EF is shorter than corridor AB

Answer: C

Q3. If another corridor GH is drawn perpendicular to corridor CD at point P, what is the relationship between corridors AB and GH?

  • A. They are parallel to each other ✓
  • B. They are perpendicular to each other
  • C. They intersect at 45°
  • D. They are on different planes

Answer: A

Q4. The design team needs a corridor that intersects both AB and CD but is neither parallel nor perpendicular to either. Is this possible?

  • A. No, because any line through point P must be either parallel or perpendicular
  • B. Yes, the corridor can intersect at any angle other than 0°, 90°, or 180° ✓
  • C. No, because parallel corridors cannot intersect
  • D. Yes, but only if it doesn't pass through point P

Answer: B

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