Intersecting, parallel and perpendicular lines
Railway tracks never meet, however far they go. Two roads crossing at a chowk meet at one point. A lamp post stands straight on the road, making a right angle.
Two lines in a plane that meet at one point are intersecting lines. Lines that never meet, always at the same distance, are parallel lines. If intersecting lines make a 90° angle, they are perpendicular lines.
In the figure: Parallel lines never meet; intersecting lines meet at a point; perpendicular lines meet at 90°.
Rule 1: Parallel lines — They are always the same distance apart and never meet.
- Easy: Opposite edges of a ruler.
- Medium: Two rails of a railway track.
- Exam-type: The top and bottom edges of a notebook page.
Rule 2: Intersecting lines — They share exactly one common point, called the point of intersection.
- Easy: Two roads at a chowk.
- Medium: The two blades of an open pair of scissors.
- Exam-type: The letter X.
Rule 3: Perpendicular lines — Intersecting lines forming a right angle (90°) are perpendicular; mark it with a small square.
- Easy: Adjacent edges of a door frame.
- Medium: The plus sign +.
- Exam-type: A flagpole standing straight on level ground.
Perpendicular lines are a special case of intersecting lines.
Thinking lines that do not meet on the page are parallel. Lines meeting if extended are not parallel.
Parallel = same gap everywhere. Check the gap at two places. Memory clue: "Parallel = rails, Perpendicular = plus."
Spot the difference
| Point | Parallel | Intersecting | Perpendicular |
|---|---|---|---|
| Meet? | never | at 1 point | at 1 point |
| Angle | none | any angle | 90° |
Practice questions
- 1Name two parallel things in your classroom.
- 2Two lines meet at exactly one point. What are they called?
- 3Intersecting lines make a 90° angle. What are they called?
- 4Are perpendicular lines also intersecting lines?
- 5How many common points do two parallel lines have?
- 6How many common points do two intersecting lines have?
- 7Two lines meet if extended on a page. Are they parallel?
- 8What angle do perpendicular lines make?
Show answers (try them yourself first)
- 1e.g. opposite edges of a ruler, rails
- 2intersecting lines
- 3perpendicular lines
- 4Yes
- 50
- 61
- 7No
- 890°
Vertically opposite angles and linear pair
When scissors open, the two opposite gaps look equal, and the gaps side by side together make a straight edge.
Two intersecting lines make four angles. Angles opposite each other are vertically opposite angles and are equal. Two angles side by side on a straight line are a linear pair and add up to 180°.
In the figure: Opposite angles are equal (60° and 60°, 120° and 120°); neighbours add to 180°.
Vertically opposite angles and linear pair
In the app you can move this figure with your finger.
Open in the appRule 1: Vertically opposite angles are equal — If one angle is known, the angle opposite to it is the same.
- Easy: One angle is 70°; the opposite angle is 70°.
- Medium: One angle is 115°; the opposite angle is 115°.
- Exam-type: If an angle is x and its opposite is 48°, then x = 48°.
Rule 2: Linear pair adds to 180° — Subtract the known angle from 180° to get its neighbour.
- Easy: One angle 70° → neighbour 180° – 70° = 110°.
- Medium: One angle 125° → neighbour 55°.
- Exam-type: Two neighbours are 2x and 3x on a line: 5x = 180°, x = 36°.
Rule 3: All four angles — Find one neighbour with 180°, then copy to the opposite angles.
- Easy: If one angle is 40°, the four are 40°, 140°, 40°, 140°.
- Medium: If one angle is 90°, all four are 90° (perpendicular).
- Exam-type: The four angles always add up to 360°.
Opposite = equal. Neighbour = 180° minus.
Writing the opposite angle as 180° – 70° = 110°. The opposite angle is 70°; 110° is the neighbour.
Two choices only: same (opposite) or 180° minus (neighbour). Memory clue: "Opposite Same, Neighbour Sum-to-180."
Spot the difference
| Point | Vertically opposite | Linear pair |
|---|---|---|
| Position | opposite each other | side by side on a line |
| Relation | equal | sum 180° |
Practice questions
- 1One of two crossing lines makes 70° with the other. Find the opposite angle.
- 2Find the neighbour of 70° on a line.
- 3Find the neighbour of 125° on a line.
- 4Two lines cross; one angle is 40°. Write all four angles.
- 5Opposite angles are x and 48°. Find x.
- 6Two angles on a line are 2x and 3x. Find x.
- 7Four angles at a crossing: one is 90°. What are the others?
- 8Neighbouring angles are (3a) and (2a + 20)°. Find a.
Show answers (try them yourself first)
- 170°
- 2110°
- 355°
- 440°, 140°, 40°, 140°
- 548°
- 636°
- 7all 90°
- 8a = 32 (5a + 20 = 180)
Transversal and corresponding angles
A road cuts across two parallel railway tracks. At each crossing the road makes the same angle with the track.
A line that crosses two (or more) lines is a transversal. It makes 8 angles. Angles in the same position at the two crossings are corresponding angles. If the two lines are parallel, corresponding angles are equal.
In the figure: The two 50° angles (top-right at each crossing) are corresponding and equal.
Transversal and corresponding angles
In the app you can move this figure with your finger.
Open in the appRule 1: Find corresponding angles — Same corner at each crossing: top-left with top-left, bottom-right with bottom-right.
- Easy: Top-left at the upper line matches top-left at the lower line.
- Medium: There are 4 pairs of corresponding angles.
- Exam-type: Only parallel lines make them equal.
Rule 2: Use equality — With parallel lines, copy the angle to its corresponding place.
- Easy: If one angle is 50°, its corresponding angle is 50°.
- Medium: If one angle is 130°, its corresponding angle is 130°.
- Exam-type: Corresponding angles x + 20 and 75: x = 55.
Rule 3: Test for parallel lines — If corresponding angles are equal, the two lines are parallel.
- Easy: Corresponding angles 65° and 65° → parallel.
- Medium: Corresponding angles 65° and 70° → not parallel.
- Exam-type: Used to check lines drawn with a set square.
"F-shape": corresponding angles sit in the corners of the letter F.
Taking corresponding angles as equal when the lines are not parallel.
First check the arrows ▸ showing parallel lines, then use equal. Memory clue: "F-angles are Fair (equal)."
Spot the difference
| Point | Lines parallel | Lines not parallel |
|---|---|---|
| Corresponding angles | equal | not equal |
| Use | find unknown angles | only measure |
Practice questions
- 1How many angles does a transversal make with two lines?
- 2One of two corresponding angles is 50° (lines parallel). Find the other.
- 3How many pairs of corresponding angles are there?
- 4A corresponding angle is 130°. Find its partner.
- 5Corresponding angles x + 20 and 75 with parallel lines. Find x.
- 6Corresponding angles are 65° and 65°. Are the lines parallel?
- 7Corresponding angles are 65° and 70°. Are the lines parallel?
- 8Corresponding angles are 3y and 2y + 18 (parallel). Find y.
Show answers (try them yourself first)
- 18
- 250°
- 34
- 4130°
- 555
- 6Yes
- 7No
- 818
Alternate and co-interior angles
Between two rails, a cross-plank makes a Z-shape on one side and a C-shape on the other. These shapes hide two more angle pairs.
Angles between the two lines and on opposite sides of the transversal are alternate interior angles (Z-shape); they are equal for parallel lines. Angles between the lines on the same side are co-interior angles (C-shape); they add up to 180°.
In the figure: Z-shape: the two green 50° angles are alternate and equal.
In the figure: C-shape: 130° + 50° = 180° (red angles) are co-interior.
Rule 1: Alternate angles (Z) — Trace a Z; the two inner corners are equal when the lines are parallel.
- Easy: One alternate angle is 50° → the other is 50°.
- Medium: One is 130° → the other is 130°.
- Exam-type: Alternate angles 2x + 10 and 70: x = 30.
Rule 2: Co-interior angles (C) — Trace a C; the two inner angles add to 180° for parallel lines.
- Easy: One angle 110° → the other 70°.
- Medium: One angle 50° → the other 130°.
- Exam-type: Co-interior angles 3x and 2x: 5x = 180°, x = 36°.
Rule 3: Which rule to use — Same position → equal (corresponding). Z-shape → equal (alternate). C-shape → 180°.
- Easy: F → equal.
- Medium: Z → equal.
- Exam-type: C → add to 180°.
Z and F give equal angles. C gives 180°.
Using "equal" for co-interior angles. 130° and 50° are not equal; they add to 180°.
Draw the letter on the figure and name it Z, F or C. Memory clue: "Z and F are Friends (equal); C is Complete (180°)."
Spot the difference
| Pair | Shape | Relation (parallel lines) |
|---|---|---|
| Corresponding | F | equal |
| Alternate | Z | equal |
| Co-interior | C | sum 180° |
Practice questions
- 1One alternate angle is 50° (parallel lines). Find the other.
- 2One co-interior angle is 110°. Find the other.
- 3Which letter shape shows alternate angles?
- 4One co-interior angle is 50°. Find the other.
- 5Alternate angles 2x + 10 and 70. Find x.
- 6Co-interior angles 3x and 2x. Find x.
- 7Co-interior angles are (x + 40)° and (2x + 20)°. Find x.
- 8Alternate angles 4y – 5 and 3y + 15. Find y.
Show answers (try them yourself first)
- 150°
- 270°
- 3Z
- 4130°
- 530
- 636
- 740 (3x + 60 = 180, x = 40)
- 820
Finding unknown angles and drawing parallel lines
A carpenter slides a set square along a ruler to draw lines of equal slope. The lines never meet: they are parallel.
To find an unknown angle, use the angle rules step by step: opposite angles are equal, a linear pair adds to 180°, corresponding and alternate angles are equal, co-interior angles add to 180° (when lines are parallel). To draw parallel lines, slide a set square along a ruler and draw along the same edge.
In the figure: The corresponding angle of 65° on the lower line is also 65°; x is its neighbour, so x = 180° – 65° = 115°.
In the figure: Slide the set square along the ruler: both edges drawn are parallel.
Rule 1: Step-by-step angle chase — Write each angle you find on the figure; each one helps find the next.
- Easy: Parallel lines, angle 65°: its corresponding angle is 65°, its neighbour is 115°.
- Medium: Opposite angle of 115° is 115°.
- Exam-type: Co-interior of 115° is 65°.
Rule 2: Forming equations — If angles are in letters, make an equation using the rule and solve.
- Easy: Corresponding angles x + 20 and 75: x = 55.
- Medium: Linear pair 3a and 2a + 20: 5a = 160, a = 32.
- Exam-type: Co-interior x + 40 and 2x + 20: 3x = 120, x = 40.
Rule 3: Drawing parallel lines — Hold the ruler fixed, slide the set square, draw along the same edge each time.
- Easy: Draw line 1 along the set-square edge.
- Medium: Slide it along the ruler to a new place and draw line 2.
- Exam-type: Check: equal corresponding angles with any transversal.
Mark every angle you find on the figure before moving on.
Using Z/F/C rules when the lines are not parallel. Check the ▸ marks first.
Go step by step: opposite → neighbour → corresponding. Memory clue: "Mark, Name, Calculate": mark angles, name the letter, calculate.
Spot the difference
| Angle pair | Parallel lines needed? | Result |
|---|---|---|
| Opposite | no | equal |
| Linear pair | no | 180° |
| F, Z, C pairs | yes | equal / 180° |
Practice questions
- 1Parallel lines: an angle is 65°. Find its corresponding angle.
- 2Parallel lines: find the neighbour of 65°.
- 3Find the co-interior angle of 115° (parallel lines).
- 4Find x if 3a and 2a + 20 form a linear pair, a = ?
- 5What tool pair is used to draw parallel lines?
- 6How do you check that two drawn lines are parallel?
- 7Parallel lines: an angle is 40°. Write its co-interior and corresponding partners.
- 8Co-interior angles are (x + 40)° and (2x + 20)°. Find x.
Show answers (try them yourself first)
- 165°
- 2115°
- 365°
- 432
- 5ruler and set square
- 6corresponding angles are equal
- 7co-interior 140°, corresponding 40°
- 840