Question

In the given figure, measures of some angles are shown.

Using the measures find the measures of $\angle$x and $\angle$y and hence show that line l || line m.

Answer 1

Suppose n is a transversal of the given lines l and m.
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 intersects line l and line m at K and L respectively.

Since PQ is a straight line and ray KA stands on it, then
∠AKL + ∠AKP = 180^∘ (angles in a linear pair)
⇒∠x + 130^∘ = 180^∘
⇒∠x = 180^∘ − 130^∘ = 50^∘

Since CD is a straight line and ray LK stands on it, then
∠KLC + ∠KLD = 180^∘ (angles in a linear pair)
⇒∠y + 50^∘ = 180^∘
⇒∠y = 180^∘ − 50^∘ = 130^∘
Now, ∠x + ∠y = 50^∘ + 130^∘ = 180^∘
But ∠x and ∠y are interior angles formed by a transversal n of line l and line m.
It is known that, if the sum of the interior angles formed by a transversal of two distinct lines is 180^∘, then the lines are parallel.
∴ line l || line m.