Question

draw an angle bisector between any opposite vertices of the hexagon so that it forms a transversal between opposite sides of the hexagon. the lines containing opposite sides are so the opposite sides are. the interior angles of a regular hexagon are congruent. why are any pair of opposite sides parallel? the angles on the opposite sides of the transversal are so the opposite sides are. video textbook get more help - clear all

Explanation:

Step1: Recall properties of a regular hexagon

A regular hexagon has 6 equal - sided and 6 equal - angled vertices. The interior angle of a regular hexagon is calculated by the formula $\theta=\frac{(n - 2)\times180^{\circ}}{n}$, where $n = 6$. So, $\theta=\frac{(6 - 2)\times180^{\circ}}{6}=120^{\circ}$.

Step2: Draw the transversal

Draw a line (angle - bisector) connecting opposite vertices of the hexagon. This line acts as a transversal between opposite sides.

Step3: Analyze the angles formed

When we draw the transversal between opposite sides of a regular hexagon, the alternate - interior angles formed are equal. For example, if we consider two opposite sides and the transversal connecting their opposite vertices, the alternate - interior angles are both $60^{\circ}$ (since the interior angle of a regular hexagon is $120^{\circ}$ and the angle - bisector divides it into two $60^{\circ}$ angles).

Step4: Apply parallel - line criteria

According to the alternate - interior angles theorem, if two lines are cut by a transversal and the alternate - interior angles are equal, then the two lines are parallel. So, the opposite sides of a regular hexagon are parallel.

Answer:

The opposite sides of a regular hexagon are parallel because when a transversal (line connecting opposite vertices) is drawn between opposite sides, the alternate - interior angles formed are equal.