Lines and Angles
CBSE Class 9 · Mathematics · Notes, formulas and practice questions
Revision notes for the CBSE Class 9 chapter on lines and angles: angle pairs, parallel lines and a transversal, and the two angle properties of triangles.
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The chapter starts with the basic definitions of points, rays, lines and angles, and then looks at the angles made when two lines intersect. It also revises complementary and supplementary angle pairs. Two key results stand out: a linear pair adds up to 180°, and vertically opposite angles are equal. These facts become the first tools for finding unknown angles in most of the chapter.
A transversal is a line that cuts two or more lines at distinct points. When the cut lines are parallel, the angle pairs formed have fixed relations: corresponding angles are equal, alternate interior angles are equal, and co-interior (or same-side) angles sum to 180°. Each relation also has a converse, so if any one of these conditions holds for two lines, the lines are parallel.
The chapter also shows that parallelism is transitive. If two lines are each parallel to a third line, then they are parallel to each other. This property works like equality: knowing that line p is parallel to line q and line q is parallel to line r immediately gives p parallel to r. It is used in diagrams containing several parallel lines, and it allows you to reduce a new line problem to a known reference line.
Finally, the parallel-line relations are used to prove two properties of triangles. The sum of the interior angles of a triangle is 180°, and when one side is extended, the exterior angle equals the sum of the two opposite interior angles. Many problems combine these triangle rules with parallel-line angle pairs, so your strategy should be to identify parallel lines, apply equal or supplementary angle pairs, and only then use the angle-sum or exterior-angle fact.
Key terms
- Linear pair of angles
- Two adjacent angles are said to form a linear pair if their non-common sides are opposite rays. The sum of the angles in a linear pair is always 180°.
- Vertically opposite angles
- When two lines intersect, the angles opposite each other at the point of intersection are called vertically opposite angles. They are always equal.
- Transversal
- A line that intersects two or more distinct lines at different points is called a transversal. When it crosses two parallel lines, several corresponding, alternate and co-interior angle pairs are formed.
- Corresponding angles
- When a transversal cuts two lines, the angles lying on the same side of the transversal and in the same relative position with respect to the two lines are called corresponding angles. If the lines are parallel, corresponding angles are equal.
- Alternate interior angles
- When a transversal cuts two lines, the pair of angles on the inner side of the two lines but on opposite sides of the transversal are alternate interior angles. If the lines are parallel, each pair is equal.
- Co-interior angles
- Also called consecutive interior angles, these are pairs of angles on the inner side of two lines but on the same side of the transversal. If the lines are parallel, each such pair sums to 180°.
- Lines parallel to the same line
- If two given lines are each parallel to a third line, then they are parallel to each other. This is a sufficient condition for parallelism.
- Angle sum property of a triangle
- The sum of the three interior angles of a triangle is always 180°. Therefore, if two angles are known, the third is found by subtracting their sum from 180°.
- Exterior angle property of a triangle
- When a side of a triangle is extended, the exterior angle formed is equal to the sum of the two opposite interior angles.
Formula sheet
| What | Formula | Notes |
|---|---|---|
| Angle sum property of a triangle | ∠A + ∠B + ∠C = 180° | ∠A, ∠B and ∠C are the three interior angles. It applies to every triangle. |
| Exterior angle property of a triangle | ∠ACD = ∠A + ∠B | In triangle ABC, side BC is extended to D; ∠ACD is the exterior angle at C, and ∠A and ∠B are its two opposite interior angles. |
| Linear pair of angles | ∠1 + ∠2 = 180° | ∠1 and ∠2 are adjacent angles whose non-common sides are opposite rays. This relation holds for any linear pair. |
| Alternate interior angles formed by a transversal on parallel lines | ∠1 = ∠2 | When two parallel lines are cut by a transversal, each pair of alternate interior angles is equal. |
| Co-interior angles formed by a transversal on parallel lines | ∠1 + ∠2 = 180° | When two parallel lines are cut by a transversal, each pair of interior angles on the same side of the transversal is supplementary. |
Practice questions with answers
1. Lines AB and CD intersect at O. If ∠AOC = 55°, find ∠BOD, ∠AOD and ∠COB.
∠BOD = ∠AOC = 55° (vertically opposite angles). ∠AOD = 180° − 55° = 125° (linear pair). ∠COB is also 125°, as it is vertically opposite to ∠AOD. Together the four angles make 360°, confirming the values.
2. An angle is 24° more than its complement. Find the angle.
Let the angle be x. Its complement is 90° − x. According to the condition, x = (90° − x) + 24°, which gives 2x = 114°, so x = 57°. The complement is 33°, and the difference is 24°.
3. Two parallel lines are cut by a transversal. If a pair of corresponding angles measures (3x + 20)° and (5x − 10)°, find x and the measure of each angle.
For parallel lines, corresponding angles are equal, so 3x + 20 = 5x − 10. Solving gives 2x = 30, hence x = 15. Each angle is 3(15) + 20 = 65°, which also equals 5(15) − 10.
4. Two parallel lines are cut by a transversal. One co-interior angle is four times the other. Find both angles.
Let the smaller angle be y and the larger be 4y. Co-interior angles on the same side are supplementary, so y + 4y = 180°, giving 5y = 180° and y = 36°. Therefore the angles are 36° and 144°.
5. In triangle ABC, ∠A = 40° and ∠B = 65°. Find ∠C.
The sum of the interior angles of a triangle is 180°. Thus ∠C = 180° − 40° − 65° = 75°.
6. An exterior angle of a triangle is 120°, and the two interior opposite angles are equal. Find all the interior angles of the triangle.
Let each opposite interior angle be x. By the exterior angle property, 2x = 120°, so x = 60°. The interior angle adjacent to the exterior angle is 180° − 120° = 60° (or 180° − 60° − 60°). Hence all three angles are 60°.
7. The angles of a triangle are in the ratio 2 : 3 : 5. Find all the angles.
Let the angles be 2x, 3x and 5x. Then 2x + 3x + 5x = 180°, so 10x = 180° and x = 18°. The angles are 36°, 54° and 90°.
8. If AB ∥ CD and CD ∥ EF, what is the relation between AB and EF? Give reason.
AB is parallel to EF. This is because two lines that are parallel to the same line are parallel to each other. Both AB and EF are parallel to CD, so they are parallel to each other.
9. Can a triangle have two right angles? Explain.
No. The three interior angles of a triangle always add to 180°. If two angles were 90° each, their sum alone would be 180°, leaving 0° for the third angle, which is impossible because every angle of a triangle has a positive measure.
10. State the relations for alternate interior angles and co-interior angles when a transversal cuts two parallel lines.
Alternate interior angles are equal to each other. Co-interior angles, which lie on the same side of the transversal, are supplementary and their measures add to 180°.
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