The sum of the measures of two alternate interior angles formed by two parallel lines with a transversal is 210 degrees. What are the possible measures of the angles formed by the parallel lines and the transversal?

Answer 1

Since alternate interior angles are congruent, let's call the measures of the two angles $x$ and $x$.

The sum of these two angles is $x+x=2x=210^\circ$. Therefore, we have $2x=210^\circ$.

Dividing both sides by $2$, we find $x=\frac{210^\circ}{2}=105^\circ$.

Therefore, the measures of the angles formed by the parallel lines and the transversal are $\boxed{105^\circ}$ for both angles.