1. Chapter at a Glance
Look at railway tracks, the lines on a notebook page, the edges of a door, a pair of scissors or a road crossing. Some lines never meet (parallel lines) and some cross each other (intersecting lines). When lines meet, they make angles that follow beautiful, exact rules. In this chapter you will discover the angle relationships formed by intersecting lines and by a line crossing two parallel lines, and learn to use them to find unknown angles.
- Intersecting lines, perpendicular lines and parallel lines.
- Linear pairs and vertically opposite angles.
- A transversal and the eight angles it makes with two lines.
- Corresponding, alternate and interior angles on the same side.
- Properties of these angles when the lines are parallel, and tests for parallel lines.
- Drawing parallel lines and solving angle problems.
2. Intersecting Lines and Linear Pairs
Two lines in a plane that meet at a point are intersecting lines; the common point is the point of intersection. Two different lines can meet at most at one point.
When two lines cross, they form four angles around the point. Any two angles next to each other (sharing an arm) and together forming a straight line are called a linear pair.
The two angles of a linear pair add up to 180°, because together they make a straight angle.
3. Vertically Opposite Angles
The angles that are opposite each other at the point of intersection (not neighbours) are called vertically opposite angles. In the figure, a and c are vertically opposite, and so are b and d.
Reason: a + b = 180° (linear pair) and b + c = 180° (linear pair). So a + b = b + c, which gives a = c. Similarly b = d.
Two lines intersect and one of the angles is 65°. Find the other three angles.
Solution: The angle next to it = 180° − 65° = 115° (linear pair). The opposite angles are equal: the four angles are 65°, 115°, 65°, 115°.
Perpendicular lines
If two intersecting lines make a 90° angle, all four angles are 90° (each is 180° − 90°). Such lines are perpendicular, written l ⊥ m.
4. Parallel Lines
Lines in the same plane that never meet, however far they are extended, are parallel lines, written l ∥ m. The distance between two parallel lines is the same everywhere. Examples: opposite edges of a ruler, rails of a straight railway track, lines on a ruled page.
Note: lines that are not in the same plane (like an edge of the ceiling and a non-parallel edge of the floor) may also never meet without being parallel — such lines are called skew lines. In this chapter we deal only with lines in one plane.
5. Transversals and the Angles They Make
A line that crosses two (or more) lines at different points is called a transversal. A transversal crossing two lines makes eight angles, four at each crossing.
| Pair type | Description | Pairs in figure |
|---|---|---|
| Corresponding angles | Same position at each crossing | 1 & 5, 2 & 6, 3 & 7, 4 & 8 |
| Alternate interior angles | Between the lines, on opposite sides of t | 3 & 5, 4 & 6 |
| Alternate exterior angles | Outside the lines, on opposite sides of t | 1 & 7, 2 & 8 |
| Interior angles on the same side (co-interior) | Between the lines, same side of t | 3 & 6, 4 & 5 |
6. Angles Formed with Parallel Lines
When the two lines cut by the transversal are parallel, these special relationships hold:
- Corresponding angles are equal (∠1 = ∠5, etc.).
- Alternate interior angles are equal (∠3 = ∠5, ∠4 = ∠6).
- Alternate exterior angles are equal (∠1 = ∠7, ∠2 = ∠8).
- Interior angles on the same side add up to 180° (∠3 + ∠6 = 180°).
Why are corresponding angles equal? Slide line l down along the transversal until it sits on line m. Because the lines are parallel, l lands exactly on m, and every angle at the top crossing lands on its matching angle at the bottom crossing.
Why are alternate angles equal? ∠3 = ∠1 (vertically opposite) and ∠1 = ∠5 (corresponding), so ∠3 = ∠5.
Tests for parallel lines (converses)
The properties also work backwards. Two lines cut by a transversal are parallel if any one of these is true:
- a pair of corresponding angles is equal;
- a pair of alternate interior angles is equal;
- a pair of interior angles on the same side adds up to 180°.
l ∥ m and a transversal makes ∠1 = 72°. Find all the other angles.
Solution: ∠2 = 180° − 72° = 108° (linear pair). ∠3 = ∠1 = 72° (vertically opposite), ∠4 = ∠2 = 108°. At the lower line, corresponding angles give ∠5 = 72°, ∠6 = 108°, ∠7 = 72°, ∠8 = 108°.
7. Drawing Parallel Lines
- Draw a line l and mark a point P not on it.
- Place one edge of a set square along l and a ruler against the other edge of the set square.
- Holding the ruler firmly, slide the set square along the ruler until its edge passes through P.
- Draw a line through P along the edge. This line is parallel to l, because the set square keeps the same (corresponding) angle with the ruler throughout.
Another method: draw a transversal through P, measure the angle it makes with l, and draw an equal corresponding angle at P.
8. Solved Problems (NCERT Style)
Two angles of a linear pair are in the ratio 4 : 5. Find them.
Solution: Let them be 4x and 5x. 9x = 180°, so x = 20°. The angles are 80° and 100°.
Two parallel lines are cut by a transversal. One interior angle on a side is 3 times the other on the same side. Find them.
Solution: They add up to 180°: x + 3x = 180°, so x = 45°. The angles are 45° and 135°.
A transversal makes alternate interior angles of 58° and 62° with lines p and q. Are p and q parallel?
Solution: For parallel lines, alternate interior angles must be equal. 58° ≠ 62°, so p and q are not parallel — they will meet on one side.
Three lines meet at one point and make six angles. Five of them are 40°, 60°, 80°, 40° and 60°. Find the sixth.
Solution: The angles around a point add up to 360°. 40 + 60 + 80 + 40 + 60 = 280°, so the sixth is 80° — which agrees with the fact that it is vertically opposite the other 80° angle.
A road runs straight, turns at A and then continues parallel to its original direction after a second turn at B. If the first turn makes an interior angle of 125° with AB, what angle does the road make with AB at B (on the same side)?
Solution: The two road segments are parallel and AB is a transversal. Interior angles on the same side add to 180°, so the angle at B is 180° − 125° = 55°.
9. Master Formula Table
| Situation | Rule |
|---|---|
| Linear pair | sum = 180° |
| Vertically opposite angles | equal |
| Angles around a point | sum = 360° |
| Perpendicular lines | all four angles 90° |
| Corresponding angles (parallel) | equal |
| Alternate interior/exterior (parallel) | equal |
| Interior angles on same side (parallel) | sum = 180° |
| Test for parallel lines | any one of the above holds |
10. Master Summary Table
| Concept | Key fact | Exam focus |
|---|---|---|
| Intersecting lines | Meet at one point | Four angles formed |
| Linear pair | Adjacent, sum 180° | Find missing angle |
| Vertically opposite | Equal | Proof using linear pairs |
| Parallel lines | Never meet, constant distance | Symbol ∥ |
| Transversal | Crosses two lines; 8 angles | Name angle pairs |
| Parallel properties | Equal or supplementary pairs | Find all 8 angles |
| Converse tests | Prove lines parallel | Check given angles |
11. Mind Map
- Linear pair = 180°
- Vertically opposite equal
- Perpendicular = 90°
- Never meet
- Same distance apart
- Corresponding
- Alternate
- Co-interior
- Find unknown angles
- Test parallelism
- Draw parallels
12. Tips and Tricks
- With parallel lines and a transversal, every angle is either equal to a given angle x or equal to 180° − x. Find one and you have all eight.
- Corresponding angles form an F shape, alternate angles a Z shape and co-interior angles a C (or U) shape.
- Mark equal angles with the same symbol on the figure as you find them.
- Using the equal-angle properties when the lines are not parallel.
- Thinking co-interior angles are equal — they add up to 180°.
- Calling any two adjacent angles a linear pair — their outer arms must form a straight line.
13. Important One-liners
- Two distinct lines can intersect in at most one point.
- Angles of a linear pair add up to 180°.
- Vertically opposite angles are always equal.
- Angles around a point add up to 360°.
- Parallel lines never meet and stay the same distance apart.
- A transversal crossing two lines makes eight angles.
- For parallel lines, corresponding angles are equal.
- For parallel lines, alternate interior angles are equal.
- For parallel lines, interior angles on the same side are supplementary.
- If corresponding angles are equal, the lines are parallel.
14. Exam-pattern Questions with Detailed Answers
One angle of a linear pair is 38°. The other is:
- 52°
- 142°
- 38°
- 322°
Show answer
Linear pair angles add to 180°: 180° − 38° = 142°.
Answer: (B) 142°
Two lines intersect, making an angle of 90°. What are the other three angles?
Show answer
The adjacent angles are 180° − 90° = 90°, and opposite angles are equal. All four are 90°.
Answer: 90°, 90°, 90°
If l ∥ m and a corresponding angle at line l is 110°, what is the matching angle at line m?
Show answer
Corresponding angles are equal for parallel lines.
Answer: 110°
Two intersecting lines make angles x and 3x − 20° that form a linear pair. Find x and all four angles.
Show answer
x + 3x − 20° = 180° → 4x = 200° → x = 50°. The other angle is 130°.
Four angles: 50°, 130°, 50°, 130°.
Answer: x = 50°; angles 50°, 130°, 50°, 130°
l ∥ m, and a transversal makes an alternate interior angle of (2y + 10)° at l and 70° at m. Find y.
Show answer
Alternate interior angles are equal: 2y + 10 = 70 → y = 30.
Answer: y = 30
Interior angles on the same side of a transversal are 95° and 85°. Are the lines parallel?
- Yes
- No
- Cannot say
- Only if one angle is 90°
Show answer
95° + 85° = 180°, which is the condition for parallel lines.
Answer: (A) Yes
Four angles are formed at a point by two intersecting lines. If the sum of two vertically opposite angles is 140°, find all four angles.
Show answer
Vertically opposite angles are equal, so each is 70°. The other two are 180° − 70° = 110° each.
Answer: 70°, 110°, 70°, 110°
Name the angle pair type: two angles between two lines, on opposite sides of the transversal.
Show answer
Between the lines = interior; opposite sides of the transversal = alternate.
Answer: Alternate interior angles
AB ∥ CD. A point E lies between the two lines. ∠ABE = 40° and ∠CDE = 35° (both measured on the same side, towards E). Find ∠BED.
Show answer
Draw a line through E parallel to AB (and hence to CD). It splits ∠BED into two parts.
The upper part equals ∠ABE = 40° (alternate angles) and the lower part equals ∠CDE = 35° (alternate angles).
∠BED = 40° + 35° = 75°.
Answer: 75°
Two parallel lines are cut by a transversal. One of the eight angles is 4 times another of them that is not equal to it. Find all the angles.
Show answer
The eight angles take only two values, x and 180° − x. If one is 4 times the other: x + 4x = 180°, so x = 36°.
Four angles are 36° and four are 144°.
Answer: 36° (four angles) and 144° (four angles)
Explain why two lines both perpendicular to the same line are parallel to each other.
Show answer
The common line acts as a transversal. Both lines make 90° with it at corresponding positions. Since the corresponding angles are equal (90° = 90°), the two lines are parallel.
Answer: Equal corresponding angles (90°) ⇒ parallel
A ladder leans against a wall, making 65° with the level ground. The wall is perpendicular to the ground. What angle does the ladder make with the wall?
Show answer
The ground, wall and ladder form a right triangle. The angles add to 180°: 90° + 65° + angle with wall = 180°, so the angle with the wall = 25°.
Answer: 25°
15. Quick Revision Summary
- Intersecting lines: linear pairs add to 180°; vertically opposite angles are equal; angles at a point total 360°.
- Parallel lines never meet; a transversal crossing them makes eight angles.
- For parallel lines: corresponding equal (F), alternate equal (Z), co-interior add to 180° (C).
- Converse: any one of these conditions proves two lines are parallel.
- Draw parallels by sliding a set square along a ruler.