Question

which of the following explains how \\( \triangle bei \\) could be proven similar to \\( \triangle ceh \\) using the aa similarity postulate?\\( \angle bei \cong \angle ceh \\) because vertical angles are congruent; rotate \\( \triangle ceh \\) 180° around point e, then translate point c to point b to confirm \\( \angle ibe \cong \angle hce \\).\\( \angle bei \cong \angle ceh \\) because vertical angles are congruent; reflect \\( \triangle ceh \\) across segment fg, then translate point c to point b to confirm \\( \angle ibe \cong \angle hce \\).\\( \angle bei \cong \angle ceh \\) because vertical angles are congruent; rotate \\( \triangle ceh \\) 180° around point e, then dilate \\( \triangle ceh \\) to confirm \\( \overline{eb} \cong \overline{ec} \\).\\( \angle bei \cong \angle ceh \\) because vertical angles are congruent; reflect \\( \triangle ceh \\) across segment fg, then dilate \\( \triangle ceh \\) to confirm \\( \overline{eh} \cong \overline{ei} \\).

Explanation:

Step1: Recall vertical angles property

Vertical angles are congruent. So, \(\angle BEI\cong\angle CEH\) as they are vertical angles.

Step2: Analyze rotation

Rotating \(\triangle CEH\) \(180^{\circ}\) around point \(E\) maps \(C\) to \(B\) and \(H\) to \(I\). After rotation, we need to check the sides. If we rotate \(\triangle CEH\) \(180^{\circ}\) around point \(E\), then \(\overline{EB}\cong\overline{EC}\) (since rotation preserves length).

Step3: Check similarity conditions

For AA (Angle - Angle) similarity postulate, we already have \(\angle BEI\cong\angle CEH\) (vertical angles). If we can show another pair of angles or the ratio of sides. Rotating \(\triangle CEH\) \(180^{\circ}\) around point \(E\) gives \(\overline{EB}\cong\overline{EC}\) which helps in establishing the similarity.

Answer:

\(\angle BEI\cong\angle CEH\) because vertical angles are congruent; rotate \(\triangle CEH\ 180^{\circ}\) around point \(E\), then \(\overline{EB}\cong\overline{EC}\)