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Lines and Angles

Lines and Angles Notes

This chapter builds the basic results about angles formed by lines: the linear pair axiom, vertically opposite angles, angles made when a transversal cuts two parallel lines (corresponding, alternate, co-interior), and the angle sum property of a triangle with its exterior angle.

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  • 12 practice questions
  • Aligned to CBSE 2025–26 syllabus
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  • Updated Sept 2026
Chapter roadmap

Use this map for a first read, then return to the checks and formulas during revision.

  1. 1.Basic Terms and the Linear Pair
  2. 2.Parallel Lines and a Transversal
  3. 3.Angle Sum and Exterior Angle of a Triangle
  • 3 key formulas
  • 1 concept checks

Basic Terms and the Linear Pair

Two angles are complementary if they add to 90° and supplementary if they add to 180°. Two angles are adjacent if they share a vertex and an arm and their other arms lie on opposite sides.

Linear pair axiom: if a ray stands on a line, the sum of the two adjacent angles so formed is 180°. Conversely, if two adjacent angles add to 180°, their outer arms form a straight line.

When two lines intersect, the vertically opposite angles are equal. This follows from applying the linear pair axiom twice.

∠AOC+∠BOC=180∘\angle AOC + \angle BOC = 180^{\circ}
Linear pair: ray OC stands on line AB.

Parallel Lines and a Transversal

A transversal is a line that cuts two or more lines at distinct points. It creates eight angles, grouped as corresponding angles, alternate interior angles, alternate exterior angles, and interior angles on the same side of the transversal (co-interior or allied angles).

If a transversal cuts two parallel lines: each pair of corresponding angles is equal, each pair of alternate interior angles is equal, and each pair of co-interior angles is supplementary (adds to 180°).

The converses are also true and are used to prove lines parallel: if a transversal makes a pair of equal corresponding angles, or equal alternate angles, or co-interior angles that add to 180°, then the two lines are parallel. Also, two lines parallel to the same line are parallel to each other.

?Check your understanding 1
A transversal cuts two parallel lines. One interior angle on a certain side of the transversal is 65°65°. The co-interior angle on the same side is:

Angle Sum and Exterior Angle of a Triangle

The angles of a triangle add up to 180°. A standard proof draws a line through one vertex parallel to the opposite side and uses alternate angles.

If one side of a triangle is produced, the exterior angle so formed equals the sum of the two interior opposite angles. Equivalently, an exterior angle is greater than either of the interior opposite angles.

Example: if a triangle has angles 50° and 60°, the third angle is 70°, and the exterior angle at the third vertex is 50° + 60° = 110°.

∠A+∠B+∠C=180∘\angle A + \angle B + \angle C = 180^{\circ}
Exterior angle=sum of the two interior opposite angles\text{Exterior angle} = \text{sum of the two interior opposite angles}

Lines and Angles: revision sheet

Close the notes and explain each idea in your own words. Then check the formulas and attempt the examples before opening their solutions.

Recall checklist
  • Basic Terms and the Linear Pair
  • Parallel Lines and a Transversal
  • Angle Sum and Exterior Angle of a Triangle

For each topic: state the rule, explain when it applies, give an example and identify one mistake to avoid. Return to any topic you cannot explain without looking.

Formula recap (3)

Basic Terms and the Linear Pair

∠AOC+∠BOC=180∘\angle AOC + \angle BOC = 180^{\circ}
Linear pair: ray OC stands on line AB.

Angle Sum and Exterior Angle of a Triangle

∠A+∠B+∠C=180∘\angle A + \angle B + \angle C = 180^{\circ}

Angle Sum and Exterior Angle of a Triangle

Exterior angle=sum of the two interior opposite angles\text{Exterior angle} = \text{sum of the two interior opposite angles}

Explain your method

Write the given information, choose a method, show the steps, then check units and whether the answer makes sense.

Two adjacent angles on a straight line are (2x + 10)° and (3x − 30)°. Find x and both angles.
Check worked solution 1
(2x+10)+(3x−30)=180⇒5x−20=180⇒x=40(2x + 10) + (3x - 30) = 180 \Rightarrow 5x - 20 = 180 \Rightarrow x = 40. Angles: 90° and 90°.
If two lines intersect and one of the angles is 130°, find the other three angles.
Check worked solution 2
Its linear pair is 50°, the vertically opposite angle is 130°, and the remaining one is 50°. Angles: 130°, 50°, 130°, 50°.

Turn mistakes into revision

Record the question, the step that went wrong and the corrected reason. Try it again tomorrow without the solution, then revisit it after a week. A correct guess still needs an explanation.

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Chapter Q&A

What is the difference between a linear pair and supplementary angles?

All linear pairs are supplementary (sum 180°) and are also adjacent, formed by a ray standing on a line. But two angles can be supplementary without being adjacent or forming a straight line.

Are vertically opposite angles always equal?

Yes. Whenever two straight lines intersect, each pair of vertically opposite angles is equal. This is a proved result, not an axiom.

How can the angle relations be used to prove two lines are parallel?

If a transversal makes a pair of equal corresponding angles, or equal alternate angles, or co-interior angles adding to 180°, then the two lines cut by it are parallel.

Why is an exterior angle of a triangle equal to the sum of the two interior opposite angles?

The exterior angle and its adjacent interior angle form a linear pair (sum 180°), and the three interior angles also sum to 180°. Subtracting the common interior angle gives the exterior angle equal to the sum of the other two interior angles.

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