Question

1. we can map $\triangle abc$ using a sequence of rigid transformations so that $a = d$, $b$ and $e$ are on the same ray from $d$, and $c$ and $f$ are on the same ray from $d$. show drawing.\
2. as a result of these transformations, $b$ must coincide with $e$. show drawing.\
3. as a result of these transformations, $c$ must coincide with $f$. show drawing.\

answer two questions about this proof.\

1. how did we show that the triangles were congruent?\

choose 1 answer:\
a we mapped one figure onto the other using rigid transformations.\
b we mapped one figure onto the other using a single transformation.\
c we showed that all corresponding sides had equal lengths and all corresponding angles had equal measures.\

1. what triangles did we show are congruent?\

choose 1 answer:\
a all triangles\
b triangles where 2 pairs of corresponding angles have the same measure, and the included corresponding sides have the same length\
c triangles where all 3 pairs of corresponding angles have the same measure

Explanation:

1)

To determine how the triangles were shown congruent, we analyze the options:

• Option A: The problem states a sequence of rigid transformations (translation, rotation, reflection) was used to map \( \triangle ABC \) onto the other triangle (implied \( \triangle DEF \)). Rigid transformations preserve size and shape, so mapping one figure onto the other via rigid transformations shows congruence.
• Option B: The problem mentions a "sequence" of rigid transformations, not a single one, so B is incorrect.
• Option C: The proof uses rigid transformations to map the figure, not directly showing all corresponding sides/angles equal (though that's a result of rigid transformations, the method here is mapping via rigid transformations). So A is correct.

To determine which triangles are congruent:

• Option A: The proof is about a specific pair of triangles (\( \triangle ABC \) and \( \triangle DEF \) or similar), not all triangles, so A is incorrect.
• Option B: The process of mapping via rigid transformations (and the congruence criteria) aligns with ASA (Angle - Side - Angle) or similar, where 2 pairs of corresponding angles and the included side are equal. The mapping shows the triangles with 2 pairs of corresponding angles equal and the included side equal are congruent.
• Option C: Having all 3 pairs of corresponding angles equal makes triangles similar, not necessarily congruent (unless sides are also equal). The proof is about congruence, so C is incorrect. Thus, B is correct.

Answer:

A. We mapped one figure onto the other using rigid transformations.

2)