Question

7 activity 1: if we know this, then we know that
triangle abc is congruent to triangle def
△abc ≅ △def

1. find a sequence of rigid motions that takes triangle abc to triangle def?
2. what is the image of segment bc after that transformation?
3. explain how you know these segments are congruent.
4. justify that angle abc is congruent to angle def?

notes

Explanation:

Step1: Analyze Rigid Motions

Rigid motions (translations, rotations, reflections) preserve shape and size. To map \( \triangle ABC \) to \( \triangle DEF \), first identify a translation to align a vertex, then a rotation/reflection. For example, translate \( A \) to \( D \), then rotate/reflection to match sides.

Step2: Image of \( \overline{BC} \)

By congruence, rigid motions map segments to congruent segments. So the image of \( \overline{BC} \) is \( \overline{EF} \) (since \( \triangle ABC \cong \triangle DEF \), corresponding parts match).

Step3: Congruence of Segments

Rigid motions (used in congruence) preserve length. Since \( \triangle ABC \cong \triangle DEF \), \( \overline{BC} \) and \( \overline{EF} \) are corresponding sides, so \( \overline{BC} \cong \overline{EF} \) (CPCTC: Corresponding Parts of Congruent Triangles are Congruent).

Step4: Congruence of Angles

\( \angle ABC \) and \( \angle DEF \) are corresponding angles of congruent triangles. By CPCTC, corresponding angles of congruent triangles are congruent, so \( \angle ABC \cong \angle DEF \).

Answer:

1. A possible sequence: Translate \( \triangle ABC \) so that \( A \) maps to \( D \), then rotate (or reflect) about \( D \) to align \( AB \) with \( DE \) and \( AC \) with \( DF \).
2. The image of \( \overline{BC} \) is \( \overline{EF} \).
3. \( \overline{BC} \cong \overline{EF} \) because rigid motions (used in triangle congruence) preserve segment length, and \( \triangle ABC \cong \triangle DEF \) implies corresponding sides are congruent (CPCTC).
4. \( \angle ABC \cong \angle DEF \) because they are corresponding angles of congruent triangles \( \triangle ABC \) and \( \triangle DEF \), and corresponding parts of congruent triangles are congruent (CPCTC).