Alternate Angles

Class 7 Maths · Chapter 5: Parallel and Intersecting Lines · NCERT Ganita Prakash section 5.8–5.9

Learn

Between the two lines and on opposite sides of the transversal are alternate angles: ∠3 and ∠5, and ∠4 and ∠6. To reach the alternate of ∠6, go to its corresponding angle ∠2, then to ∠2's vertically opposite angle ∠4. Both steps keep the size, so ∠6 = ∠4.

∠6 = ∠2 = ∠4

Alternate angles made by a transversal on parallel lines are always equal. Example: l ∥ m and ∠6 = 135°. Then ∠2, ∠4, ∠8 are 135° (corresponding and vertically opposite), and ∠1, ∠3, ∠5, ∠7 are 180° − 135° = 45°.

Interior angles on the same side, like ∠3 and ∠6, are between the lines on the same side of t. With parallel lines they add up to 180°: if ∠3 = 50°, ∠2 = 130° (linear pair) and ∠6 = ∠2 = 130°, and 50° + 130° = 180°. And if ∠2 = 60° but ∠6 = 70°, the lines are not parallel.

∠3 + ∠6 = 180°

A bent path: if l ∥ m and a path bends at a point P between them, draw a line through P parallel to both. The bend splits into two alternate angles, so ∠P = the two angles added. For 40° at l and 30° at m, ∠P = 70°. And some drawings (illusions) trick the eye: lines that are truly parallel can look bent!

∠P = 40° + 30° = 70°

Explore: alternate and interior angles.

1. Turn m until the alternate angles ∠3 and ∠5 are equal.
2. Now with interior angles: turn m until ∠3 + ∠6 = 180°.

Remember

• With parallel lines, alternate angles are equal (∠3 = ∠5, ∠4 = ∠6).
• Interior angles on the same side add up to 180°.

Quick check

l ∥ m and ∠4 = 72°. What is its alternate angle ∠6?

1. 72°
2. 108°
3. 18°
4. 144°

Think about it

ABCD has AB ∥ DC and AD ∥ BC, with ∠D = 60°. What are ∠A, ∠B and ∠C?

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.