CBSE Mathematics 2026–27 · Topic 8

Lines and Angles

Understand the idea, practise the reasoning and apply it in an investigation. These original notes follow the regular Class IX Mathematics syllabus; the Advanced course is separate.

What to learn

Be able to explain the key result, choose a method that fits the question, show reasoning clearly and verify the answer.

Core concepts

A linear pair sums to 180°; vertically opposite angles are equal. Angles around a point total 360°.
With parallel lines and a transversal, corresponding and alternate interior angles are equal; co-interior angles on the same side sum to 180°.
Use only the angle rules whose conditions are established. Show the relation first, then form an equation and solve.

Angle pairs, parallel lines, and proof

An angle is formed by two rays with a common endpoint. A straight angle measures 180°; a right angle measures 90°. Two adjacent angles whose non-common arms form a straight line are a linear pair, so their measures add to 180°. Angles around one point add to 360°.

Vertically opposite angles are equal. For example, if one is x and an adjacent angle is y, then x+y=180°. The angle vertically opposite x also forms a linear pair with y, so it too equals 180°−y=x. This short proof shows the result follows from the linear-pair fact.

When parallel lines are cut by a transversal, corresponding angles and alternate interior angles are equal; interior angles on the same side of the transversal are supplementary. The converses are useful too: if a corresponding pair is equal, or an alternate interior pair is equal, the two lines are parallel. State whether the lines are known parallel or whether an angle equality is being used to prove parallelism.

Example: If parallel lines give alternate interior angles 4x−5° and 3x+20°, set them equal: 4x−5=3x+20, so x=25. Each angle is 95°. The other angle pairs are found using equality of vertical angles or a 180° linear pair.

Triangle angles provide a useful check: the interior angles total 180°, so at least two are acute; two obtuse angles are impossible. If all three were less than 60°, their sum would be less than 180°, also impossible. A contradiction proof can establish such claims by assuming the impossible angle arrangement and comparing its sum with 180°.

Original CBSE-style solved examples

Attempt each calculation before reading the solution.

Example · corresponding angles

Parallel lines are cut by a transversal. Corresponding angles are (3x+10)° and (5x−20)°. Set equal: 3x+10=5x−20, so x=15 and each angle is 55°.

Practice questions with answers

1. One angle of a linear pair is 68°. Find the other.

112°.

2. One vertically opposite angle is 127°. Find its opposite.

127°.

Practical investigation

Draw parallel lines and a transversal. Label all eight angles, colour equal angles alike, and explain one equal pair and one supplementary pair.

Keep a record of materials, steps, observations and conclusion.

Self-check

Can I state the definition or theorem precisely?
Can I explain why the method applies?
Did I check signs, units, labels and the final result?

Use your prescribed NCERT textbook and teacher’s sequence. These are original study notes, not copied textbook answers.