Parallel Lines

👉 Parallel Lines

Two lines lying in the same plane which do not intersect each other are called parallel lines.
If the perpendicular distance between any two points of one line and the corresponding points of the other is equal, they are parallel.
Two parallel straight lines never intersect each other.

AB || CD

Transversal of Parallel Lines

MN is a transversal of AB || CD

👉 Corresponding Angles

When a straight line intersects two parallel lines obliquely, the angles formed on the same side of the transversal with the parallel lines are called corresponding angles. Corresponding angles are equal.

Corresponding Angles

∠MOB corresponds to ∠OPD

Here, the angles of the same color are corresponding angles.

👉 Alternate Angles

When a straight line intersects two parallel lines obliquely, the angles formed on the opposite sides of the transversal with the parallel lines are called alternate angles. Alternate angles are equal.

Alternate Angles

∠AOP is alternate to ∠DPO

∠BOP is alternate to ∠CPO

Proof that alternate angles are equal.+

∵	∠MOB = Corresponding ∠OPD
Again,	∠MOB = Vertically opposite ∠AOP
∴	∠AOP = Alternate ∠OPD
∴	Similarly, it can be proved that
	∠BOP = Alternate ∠CPO

👉 Internal Angles

When a straight line intersects two parallel lines obliquely, the sum of the interior angles on the same side of the transversal is always 180° or equal to two right angles.

Internal Angles

∠BOP + ∠OPD = 180°, ∠CPO + ∠AOP = 180°

Here, the angles of the same color are interior angles. The sum of two interior angles is equal to two right angles or 180°.

Proof that the sum of the interior angles is always 180°.+

∵	∠MOB = Corresponding ∠OPD
Again,	∠MOB = Vertically opposite ∠AOP
∴	∠AOP + ∠OPD = 180°
∴	Therefore, the sum of the interior angles is 180°.
∴	Similarly, it can be proved for other pairs of interior angles as well.