Question

trapezoid abcd is rotated and then translated to produce trapezoid efgh. select all the true statements. (angle f) has the same measure as (angle a). (overline{eh}) is the same length as (overline{bc}). trapezoid efgh is congruent to trapezoid abcd. (angle g) has the same measure as (angle c). (overline{ef}) is parallel to (overline{hg}).

Explanation:

Step1: Recall properties of rotation and translation

Rotation and translation are rigid - motions. Rigid - motions preserve shape and size, which means congruence. Corresponding angles and corresponding sides of congruent figures are equal.

Step2: Analyze corresponding angles

Since trapezoid \(ABCD\) is rotated and translated to produce trapezoid \(EFGH\), \(\angle F\) corresponds to \(\angle A\), so \(m\angle F=m\angle A\). Also, \(\angle G\) corresponds to \(\angle C\), so \(m\angle G = m\angle C\).

Step3: Analyze corresponding sides

\(\overline{EH}\) does not correspond to \(\overline{BC}\), so they are not necessarily the same length.

Step4: Analyze congruence

Since rotation and translation are rigid - motions, trapezoid \(EFGH\) is congruent to trapezoid \(ABCD\).

Step5: Analyze parallel sides

In trapezoid \(ABCD\), assume \(AB\parallel DC\). After rotation and translation, the parallel - side relationship is not necessarily the same as the original. In trapezoid \(EFGH\), we cannot assume \(\overline{EF}\parallel\overline{HG}\) based on the given transformation of rotation and translation from trapezoid \(ABCD\).

Answer:

\(\angle F\) has the same measure as \(\angle A\); Trapezoid \(EFGH\) is congruent to trapezoid \(ABCD\); \(\angle G\) has the same measure as \(\angle C\)