Question

problem 7
\\( \overline { a b } \cong \overline { e f } \\)
\\( \angle f \cong \angle b \\)
use rigid motions to prove that figure \\( a b c \\) is congruent to figure \\( e f g \\).

Explanation:

Step1: Identify Corresponding Parts

We know $\overline{AB} \cong \overline{EF}$, $\angle F \cong \angle B$, and from the diagram, $\overline{BC} \cong \overline{FG}$ (marked with congruency ticks).

Step2: Apply Rigid Motions - Translation

First, translate figure \( ABC \) so that point \( B \) maps to point \( F \). Since translation preserves distance and angle, \( \overline{AB} \) will align with \( \overline{EF} \) (as \( \overline{AB} \cong \overline{EF} \)) and \( \angle B \) will align with \( \angle F \) (as \( \angle B \cong \angle F \)).

Step3: Apply Rigid Motions - Rotation (if needed)

After translation, since \( \overline{BC} \cong \overline{FG} \) and \( \angle B \cong \angle F \), a rotation (around point \( F \)) can be applied to make \( \overline{BC} \) align with \( \overline{FG} \). Since rigid motions (translation + rotation) preserve congruence, point \( C \) will map to point \( G \), and point \( A \) maps to point \( E \), \( B \) to \( F \), \( C \) to \( G \). Thus, \( \triangle ABC \cong \triangle EFG \) by rigid motion (translation and rotation) which preserves congruence.

Answer:

To prove \( \triangle ABC \cong \triangle EFG \) using rigid motions:

1. Translate \( \triangle ABC \) so that \( B \) maps to \( F \). Since \( \overline{AB} \cong \overline{EF} \), \( A \) maps to \( E \) after translation.
2. Rotate (around \( F \)) \( \triangle ABC \) (now with \( B \) at \( F \)) so that \( \overline{BC} \) aligns with \( \overline{FG} \) (since \( \overline{BC} \cong \overline{FG} \) and \( \angle B \cong \angle F \)).
3. After translation and rotation (rigid motions), \( C \) maps to \( G \). Since rigid motions preserve congruence, \( \triangle ABC \cong \triangle EFG \).