Question

leah wants to prove that the angle - side - angle (asa) criterion is sufficient to show that \\( \triangle efg\cong\triangle hij \\).
leah starts by translating \\( \triangle efg\\) to get \\( \triangle efg\\), where \\( g = j \\). then she rotates \\( \triangle efg\\) to get \\( \triangle efg \\) where \\( \overrightarrow{jh} \\) and \\( \overrightarrow{ge} \\) coincide.
leah says that \\( \overrightarrow{ji} \\) and \\( \overrightarrow{gf} \\) also coincide. how can leah justify this statement?
translation and rotation preserve angle measure, so \\( m\angle egf = m\angle hij \\).
translation and rotation preserve distance, so \\( ge = jh \\).
translation and rotation preserve distance, so \\( gf = ji \\).
translation and rotation preserve angle measure, so \\( m\angle efg = m\angle hij \\).

Explanation:

Step1: Recall rigid transformations properties

Translation and rotation are rigid transformations, preserving angle measures and distances.

Step2: Link ASA to angle preservation

ASA requires two angles and the included side. After transformations, $G''=J$, $\overline{G''E''}=\overline{JH}$ (given coincidence). To have $\overline{JI}$ and $\overline{G''F''}$ coincide, the angles at $G''$ (original $\angle G$) and $J$ (original $\angle J$) must be equal, which is ensured by angle preservation.

Step3: Match with options

The option stating angle preservation leading to $m\angle E''G''F''=m\angle HJI$ justifies the coincidence of the sides.

Answer:

Translation and rotation preserve angle measure, so $m\angle E''G''F'' = m\angle HJI$.