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Two lines cross and you get four angles. A third line cuts two parallel lines and you get eight. Students often measure each one with a protractor, which is slow and, as the NCERT book admits, not always accurate. This post shows how four facts from the Class 7 chapter let you find every angle from one, with worked examples, the common slips and a practice routine.
What does the Class 7 chapter cover?
NCERT's new Class 7 book, Ganita Prakash Part 1, has Chapter 5, "Parallel and Intersecting Lines". The Part 1 contents page describes it as introducing these ideas through activities like paper folding and "rigorous mathematical reasoning". You can read the chapter itself as a free PDF: Parallel and Intersecting Lines. It covers intersecting and perpendicular lines, parallel lines, transversals, corresponding and alternate angles, and ends with some optical illusions.
What happens when two lines cross?
When two lines cross they make four angles. Call them a, b, c and d going round.
Linear pair. a and b sit side by side on a straight line, so a + b = 180°.
Vertically opposite angles. a and c face each other across the crossing point. The chapter proves they are equal without measuring: a + b = 180° and b + c = 180°, so a and c must be the same.
So if a = 60°, then b = 120°, c = 60° and d = 120°. If all four angles are equal, each is 90° and the lines are perpendicular.
What are corresponding and alternate angles?
A transversal is a line that crosses two other lines. It makes eight angles, four at each crossing.
Corresponding angles are in the same position at each crossing, such as top-right at the first and top-right at the second. The chapter's rule: if the lines are parallel, corresponding angles are equal. It also works the other way round: if corresponding angles are equal, the lines are parallel.
Alternate angles sit on opposite sides of the transversal, between the two lines. The book shows they are equal by chaining two facts: an angle equals its corresponding angle, which equals its vertically opposite angle.
Interior angles on the same side of the transversal, between the lines, add up to 180° when the lines are parallel.
Worked examples
Example 1 (from the chapter). Parallel lines are cut by a transversal and one angle is 135°. Its corresponding angle is 135°, its vertically opposite angle is 135°, and so on, so four of the eight angles are 135°. The angle beside it on the line is 180° − 135° = 45°, and the other three are 45° too. Check: 135° + 45° = 180°.
Example 2 (from the chapter). Is this pair of lines parallel? At the top crossing one angle is 120°, so its neighbour on the line is 180° − 120° = 60°. The matching corresponding angle at the bottom crossing is 70°. Since 60° is not 70°, the lines are not parallel.
Example 3 (with letter-numbers). This one previews the equation-solving you meet in "Finding the Unknown", the last chapter of Part 2. Two alternate angles on parallel lines are (3x + 10)° and (5x − 30)°. Find each angle.
Alternate angles are equal: 3x + 10 = 5x − 30.
Add 30 to both sides: 3x + 40 = 5x.
Take 3x from both sides: 40 = 2x, so x = 20.
Each angle: 3 × 20 + 10 = 70°, and 5 × 20 − 30 = 70°. They match.
Example 4. Two interior angles on the same side are (2x + 20)° and (3x + 10)°. They add to 180°: 5x + 30 = 180, so 5x = 150 and x = 30. The angles are 80° and 100°, which add to 180°.
What mistakes cost marks?
Using the parallel rules when the lines are not parallel. Corresponding angles are equal only for parallel lines. Look for the arrow marks or the word "parallel".
Mixing up alternate and same-side interior angles. Alternate angles are on opposite sides of the transversal (they make a Z shape). Same-side interior angles are on one side (a C or U shape) and add to 180°.
Trusting the drawing. The chapter notes that measured linear pairs sometimes do not add to exactly 180° because of drawing and measuring errors. Reason from the facts, then measure only to check.
Skipping the reason. Write "corresponding angles, l is parallel to m" next to each step. In tests, the reason is often worth marks.
A practice routine for the week
Day 1: Draw two crossing lines, label the angles, and write the four linear pairs and two vertically opposite pairs.
Day 2: Draw a transversal across two lines on ruled paper (the lines are parallel). Mark one angle and find the other seven with reasons.
Day 3: Do two letter-number problems like Examples 3 and 4.
Day 4: Mix it up: one angle problem, one decimals question and one expressions question. Rohrer and Taylor (2007) found that mixed practice sets gave much better results on a test a week later than practising one type at a time.
Some students find it easier to see why the rules hold by watching the angles change as a line tilts. The Fermi Simulation Lab has interactive simulations for ideas that are easier to see than to read. If you want to see where this chapter came from, revisit lines and angles in Class 6, and for where it goes next, lines and angles in Class 9.
Your next step: on ruled paper, draw one slanted line across three ruled lines, mark one angle as 50°, and find all twelve angles.
