Question

here are some statements about 2 zigzags. put them in order to prove figure abc is congruent to figure def. (lesson 2 - 5)
1: if necessary, reflect the image of figure abc across de to be sure the image of c, which we will call c’, is on the same side of de as f.
2: c’ must be on ray ef since both c’ and f are on the same side of de and make the same angle with it at e.
3: segments ab and de are the same length so they are congruent. therefore, there is a rigid motion that takes ab to de. apply that rigid motion to figure abc.
4: since points c’ and f are the same distance along the same ray from e they have to be in the same place.
5: therefore, figure abc is congruent to figure def.

Explanation:

Step1: Start with congruent segments

We first note the congruent segments. Since segments $AB$ and $DE$ are the same length, we start with the rigid - motion step.
3: Segments $AB$ and $DE$ are the same length so they are congruent. Therefore, there is a rigid motion that takes $AB$ to $DE$. Apply that rigid motion to figure $ABC$.

Step2: Adjust the position of $C$

After the initial rigid - motion, we may need to reflect to get $C$ on the same side as $F$.
1: If necessary, reflect the image of figure $ABC$ across $DE$ to be sure the image of $C$, which we will call $C'$, is on the same side of $DE$ as $F$.

Step3: Locate $C'$ on ray $EF$

Once $C'$ is on the same side of $DE$ as $F$, we use the angle - side relationship.
2: $C'$ must be on ray $EF$ since both $C'$ and $F$ are on the same side of $DE$ and make the same angle with it at $E$.

Step4: Prove $C'$ and $F$ coincide

Since $C'$ and $F$ are on the same ray and at the same distance from $E$.
4: Since points $C'$ and $F$ are the same distance along the same ray from $E$ they have to be in the same place.

Step5: Conclude congruence

Once all points coincide, we conclude the figures are congruent.
5: Therefore, figure $ABC$ is congruent to figure $DEF$.

Answer:

3, 1, 2, 4, 5