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 $$\parallel$$ 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
$$\angle AKL + \angle AKP = 180^\circ$$ (angles in a linear pair)
$$\Rightarrow \angle x + 130^\circ = 180^\circ$$
$$\Rightarrow \angle x = 180^\circ - 130^\circ = 50^\circ$$
Since CD is a straight line and ray LK stands on it, then
$$\angle KLC + \angle KLD = 180^\circ$$ (angles in a linear pair)
$$\Rightarrow \angle y + 50^\circ = 180^\circ$$
$$\Rightarrow \angle y = 180^\circ - 50^\circ = 130^\circ$$
Now, $$\angle x + \angle y = 50^\circ + 130^\circ = 180^\circ$$
But $$\angle x$$ and $$\angle y$$ 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 $$180^\circ,$$ then the lines are parallel.
$$\therefore$$ line l $$\parallel$$ line m.