Ad placement reserved for chapter sponsors, education tools, test prep platforms, and student offers.
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.
- 3 min read
- 12 practice questions
- Aligned to CBSE 2025–26 syllabus
- 100% free · no login
- Updated Aug 2026
Reserved space for student-focused ads, learning tools, scholarships, and exam prep promotions.
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.
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.
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°.
Practice and Revision
Test your understanding with quick chapter-level practice.
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.
Ad slot placed after the chapter body so reading flow is never interrupted.
This inventory appears across Class 9 and Class 10 notes so ads remain visible throughout the study journey.