Across the Line and Perpendicular Lines

Class 7 Maths · Chapter 5: Parallel and Intersecting Lines · NCERT Ganita Prakash section 5.1–5.2

Learn

Zoya folds a square of paper a few times and draws along the creases. Some lines meet at a point: we say they intersect. Two straight lines can meet at only one point. Where two lines cross, they make four angles, called a, b, c and d here.

If ∠a = 120°, then ∠a and ∠b together make a straight angle, so ∠b = 180° − 120° = 60°. In the same way ∠c = 120° and ∠d = 60°. Angles side by side like a and b are a linear pair: a linear pair always adds up to 180°.

∠a + ∠b = 180°

Why is this always true? ∠a + ∠b = 180° and ∠a + ∠d = 180°, so ∠b and ∠d must be equal, whatever ∠a is. Opposite angles like b and d are vertically opposite angles, and they are always equal. Reasoning that shows something is true every time is a proof.

∠b = ∠d · ∠a = ∠c

When you measure, a linear pair may come to 179° or 181°: drawn lines have thickness and protractors are read with small errors. Geometry reasons with ideal lines. If all four angles are equal, each is 360° ÷ 4 = 90°, and the lines are perpendicular.

4 equal angles → 90° each

Explore: linear pair or vertically opposite?

1. Sort the pairs of angles made by two crossing lines.

Remember

• Two crossing lines make 4 angles: a linear pair adds to 180°, vertically opposite angles are equal.
• Lines that cross at 90° are perpendicular.

Quick check

Two lines cross and one angle is 35°. Which list could be the four angles?

1. 35°, 145°, 35°, 145°
2. 35°, 35°, 35°, 35°
3. 35°, 55°, 35°, 55°
4. 35°, 145°, 145°, 35°

Think about it

Can two straight lines meet at two different points?

Practice

Practise with fresh questions at three levels (easy, medium, hard), some answered hands-on, with hints and worked answers. Get 4 of your last 5 right to finish the topic.