Question

the interior angles of a regular hexagon are congruent. why are any pair of opposite sides parallel? draw an angle bisector between any opposite vertices of the hexagon so that it forms a transversal between opposite sides of the hexagon. the angles on the opposite sides of the transversal are dropdown by the dropdown the lines containing opposite sides are dropdown so the opposite sides are dropdown

Explanation:

Step1: Recall regular hexagon properties

A regular hexagon has all interior angles equal to \(120^\circ\). When we draw a transversal (the angle bisector between opposite vertices), it creates alternate interior angles with the opposite sides.

Step2: Analyze alternate interior angles

For a regular hexagon, the alternate interior angles formed by the transversal with the opposite sides are equal (each \(60^\circ\) or supplementary in a way that satisfies parallel line criteria). By the Alternate Interior Angles Theorem, if alternate interior angles are congruent, the lines (sides) are parallel. So the angles on the opposite sides of the transversal are congruent, the lines containing opposite sides are parallel, so the opposite sides are parallel.

Answer:

First dropdown: (No specific value here, context is about the theorem)
Second dropdown: Alternate Interior Angles Theorem
Third dropdown: congruent
Fourth dropdown: parallel

(Note: The first dropdown might be a lead - in, but the key logical steps use the Alternate Interior Angles Theorem, note congruent alternate interior angles imply parallel lines.)