Question

In the given figure, if $\angle$a $\cong \angle$b then prove that line l || line m.

Answer 1

Let us mark the points A and B on line l, C and D on line m and P and Q on line n.
Suppose the line n intersect line l at K and line m at L.
Since PQ is a straight line and ray KA stands on it, then
m∠AKP + m∠AKL = 180^∘ (Angles in a linear pair)
⇒m∠a + m∠AKL = 180^∘
⇒m∠a = 180^∘ − m∠AKL ....(1)

Since PQ is a straight line and ray LD stands on it, then
m∠DLQ + m∠DLA = 180^∘ (Angles in a linear pair)
⇒m∠b + m∠DLA = 180^∘
⇒m∠b = 180^∘ − m∠DLA ....(2)

Since, ∠a ≅ ∠b, then m∠a = m∠b.
∴ from (1) and (2), we get
180^∘ − m∠AKL = 180^∘ − m∠DLA
⇒m∠AKL = m∠DLA
⇒∠AKL ≅ ∠DLA

It is known that, if a pair of alternate interior angles formed by a transversal of two lines is congruent, then the two lines are parallel.
∴ AB || CD or line l || line m.