CBSE Class 9 Maths Notes: Lines and Angles

Linear Pair Axiom

Definitions: A linear pair is a pair of adjacent angles formed when two lines intersect.

Core Principle: The sum of angles in a linear pair is always $180^\circ$.

Statement: If a ray stands on a line, then the sum of the two adjacent angles so formed is $180^\circ$. Conversely, if the sum of two adjacent angles is $180^\circ$, then the non-common arms of the angles form a line.

Vertically Opposite Angles Theorem

Definitions: Vertically opposite angles are the angles opposite to each other when two lines intersect.

Theorem: If two lines intersect, then the vertically opposite angles are equal.

Proof:

1. Let lines AB and CD intersect at point O.
2. We need to prove:
   • $\angle AOC = \angle BOD$
   • $\angle AOD = \angle BOC$
3. Since ray OA stands on line CD, $$\angle AOC + \angle AOD = 180^\circ \quad \text{(Linear pair axiom)} \qquad (1)$$
4. Also, since ray OD stands on line AB, $$\angle AOD + \angle BOD = 180^\circ \quad \text{(Linear pair axiom)} \qquad (2)$$
5. From equations (1) and (2), $$\angle AOC + \angle AOD = \angle AOD + \angle BOD$$
6. Subtracting $\angle AOD$ from both sides, $$\angle AOC = \angle BOD$$
7. Similarly, we can prove that $\angle AOD = \angle BOC$.
8. Hence Proved.

Parallel Lines

Definitions: Parallel lines are lines in a plane that never intersect. The distance between them remains constant.

Concept: Two lines are parallel if and only if any of the following conditions hold when intersected by a transversal:

• Corresponding angles are equal.
• Alternate interior angles are equal.
• Co-interior angles (same-side interior angles) are supplementary (sum to $180^\circ$).

Symbol: Parallel lines are denoted by “||”. For example, line AB || line CD.

Angles Made by a Transversal

Definitions: A transversal is a line that intersects two or more other lines at distinct points.

Angles Formed: When a transversal intersects two lines, eight angles are formed. These angles can be categorized as follows:

• Corresponding Angles: Angles that occupy the same relative position at each intersection. Examples: $\angle 1$ and $\angle 5$, $\angle 2$ and $\angle 6$, $\angle 3$ and $\angle 7$, $\angle 4$ and $\angle 8$.
• Alternate Interior Angles: Angles that lie between the two lines and on opposite sides of the transversal. Examples: $\angle 3$ and $\angle 6$, $\angle 4$ and $\angle 5$.
• Alternate Exterior Angles: Angles that lie outside the two lines and on opposite sides of the transversal. Examples: $\angle 1$ and $\angle 8$, $\angle 2$ and $\angle 7$.
• Interior Angles on the Same Side of the Transversal (Co-interior or Consecutive Interior): Angles that lie between the two lines and on the same side of the transversal. Examples: $\angle 3$ and $\angle 5$, $\angle 4$ and $\angle 6$.

Relationships when lines are parallel:

• Corresponding angles are equal.
• Alternate interior angles are equal.
• Alternate exterior angles are equal.
• Interior angles on the same side of the transversal are supplementary (add up to $180^\circ$).