If a transversal intersects two parallel lines, and the difference between two interior angles on the same side of the transversal is 40°, then find the smallest angle among the interior angles.

Two parallel lines are intersected by a transversal.

The difference between two interior angles on the same side of the transversal is $$40^\circ.$$​​

When a transversal intersects two parallel lines, the sum of two interior angles on the same side of the transversal is $$180^\circ$$​ (since they are supplementary).

According to the question;

x + y = 180……..(i)

x - y = 40 ………..(ii)

Add the two equations:

(x + y) + (x - y) = 180 + 40

2x = 220

x = 110

Substituting x = 110 into the first equation:

110 + y = 180

y = 70

The two angles are $$110^\circ$$​ and $$70^\circ.$$​​

Thus, The smallest angle among the interior angles is 70°.