Lines and Angles - Class 7 Mathematics (CBSE)
Based on the 2025-26 NCERT syllabus for Class 7 Mathematics. This chapter develops geometric reasoning through angle relationships, parallel lines, and transversal properties.
1. Why this chapter matters
Lines and angles form the foundation of geometry. Understanding angle relationships is essential for proving properties of triangles, quadrilaterals, and circles in higher classes. In CBSE exams, this chapter contributes 6-8 marks with diagram-based questions and angle calculations.
2. Pairs of angles
Complementary angles
Two angles whose sum is 90 degrees.
- 30 and 60 are complementary.
- If one angle is x, its complement is 90 - x.
Supplementary angles
Two angles whose sum is 180 degrees.
- 110 and 70 are supplementary.
- If one angle is x, its supplement is 180 - x.
Adjacent angles
Two angles that share:
- A common vertex
- A common arm (side)
- No common interior points
Linear pair
A pair of adjacent angles whose non-common arms form a straight line. The sum of angles in a linear pair is always 180 degrees.
Vertically opposite angles
When two lines intersect, the angles opposite each other are equal.
If lines AB and CD intersect at O:
- Angle AOC = Angle BOD (vertically opposite)
- Angle AOD = Angle BOC (vertically opposite)
3. Related angles reference table
| Angle pair | Property | Sum/Relation |
|---|---|---|
| Complementary | Sum = 90 | Each is complement of the other |
| Supplementary | Sum = 180 | Each is supplement of the other |
| Linear pair | Adjacent + straight line | Sum = 180 |
| Vertically opposite | Formed by intersecting lines | Equal |
4. Parallel lines and transversal
Definitions
- Parallel lines: Lines that never meet, no matter how far extended.
- Transversal: A line that intersects two or more lines at distinct points.
Angle pairs formed by a transversal
When a transversal cuts two lines, eight angles are formed. They are grouped as:
Corresponding angles
Angles on the same side of the transversal and on the same side of the lines. They are equal when lines are parallel.
Alternate interior angles
Angles inside the two lines but on opposite sides of the transversal. They are equal when lines are parallel.
Alternate exterior angles
Angles outside the two lines but on opposite sides of the transversal. They are equal when lines are parallel.
Interior angles on the same side of transversal
Also called co-interior or consecutive interior angles. Their sum is 180 degrees when lines are parallel.
5. Angle relationships when lines are parallel
| Angle pair | Relationship (if parallel) |
|---|---|
| Corresponding angles | Equal |
| Alternate interior angles | Equal |
| Alternate exterior angles | Equal |
| Co-interior (same-side interior) | Sum = 180 |
If any of these conditions is satisfied, the lines are parallel.
6. Checking for parallel lines
To check if two lines are parallel when a transversal cuts them, verify any one:
- A pair of corresponding angles are equal.
- A pair of alternate interior angles are equal.
- A pair of interior angles on the same side of transversal sum to 180 degrees.
7. Worked examples
Example 1: Two complementary angles are in the ratio 2:3. Find the angles.
Let the angles be 2x and 3x. 2x + 3x = 90. 5x = 90. x = 18. Angles: 36 and 54.
Example 2: Find the angle which is its own supplement.
Let angle = x. Supplement = 180 - x. If x = 180 - x, then 2x = 180. x = 90. Answer: 90 degrees.
Example 3: In the figure, lines AB and CD intersect at O. Angle AOC = 55. Find all other angles.
Vertically opposite: Angle BOD = 55. Linear pair: Angle AOD = 180 - 55 = 125. Vertically opposite: Angle BOC = 125.
Example 4: A transversal cuts two parallel lines. One interior angle is 65. Find all other angles.
Given one interior angle = 65. Corresponding angle on same side = 65. Alternate interior angle = 65. Linear pair of 65 = 115. All acute angles = 65. All obtuse angles = 115.
8. Common mistakes and how to fix them
| Mistake | Fix |
|---|---|
| Confusing complementary and supplementary | Remember: C for Corner (90), S for Straight line (180) |
| Thinking vertically opposite are adjacent | Vertically opposite angles are NOT adjacent |
| Assuming all corresponding angles are equal | They are equal ONLY when lines are parallel |
| Confusing alternate interior and exterior | Interior = between the lines; exterior = outside |
| Forgetting linear pair sum is 180 | Any linear pair always sums to 180 degrees |
9. CBSE exam focus
| Question type | Marks | Frequency |
|---|---|---|
| Complementary/supplementary angle finding | 2 marks | 1 question |
| Vertically opposite angles | 2 marks | 1 question |
| Angles in parallel lines with transversal | 3 marks | 1 question |
| Proving lines parallel | 3 marks | 1 question |
| Multiple angle finding in complex figures | 4 marks | Occasional |
10. Self-test
- Find the complement of 37 degrees.
- Find the supplement of 112 degrees.
- Two supplementary angles are in the ratio 4:5. Find the angles.
- In the figure, a transversal cuts two parallel lines. If one corresponding angle is 72, find all angles.
- Lines AB and CD intersect at O. If angle AOC : angle BOC = 2:3, find all four angles.
- State whether the following statement is true or false: 'Alternate interior angles are always equal.'
11. Answer key
- Complement = 90 - 37 = 53 degrees.
- Supplement = 180 - 112 = 68 degrees.
- 4x + 5x = 180. 9x = 180. x = 20. Angles: 80 and 100.
- All acute angles = 72. All obtuse angles = 108.
- Let angle AOC = 2x, angle BOC = 3x. 2x + 3x = 180 (linear pair). 5x = 180. x = 36. AOC = 72, BOC = 108, BOD = 72, AOD = 108.
- False. Alternate interior angles are equal only when the lines cut by the transversal are parallel.
12. Quick revision
- Complementary angles sum to 90 degrees.
- Supplementary angles sum to 180 degrees.
- Vertically opposite angles are equal.
- Linear pair sums to 180 degrees.
- Parallel lines: corresponding and alternate angles equal, co-interior sum = 180.
- Any one condition satisfied means lines are parallel.
