Question

lesson synthesis: points, segments, and zigzags
how can we prove they are congruent?

Explanation:

Step1: Identify congruence criteria

We can use Side - Angle - Side (SAS) congruence criterion.

Step2: Check side - angle - side conditions

For two triangles (assume the shapes are triangles formed by the segments), if we can show that two sides and the included angle of one triangle are equal to the corresponding two sides and included angle of the other triangle, they are congruent. For example, if in $\triangle ABC$ and $\triangle EFG$, $AB = EF$, $\angle B=\angle F$, and $BC = FG$, then $\triangle ABC\cong\triangle EFG$ by SAS.

Answer:

We can use congruence criteria like SAS (Side - Angle - Side), SSS (Side - Side - Side), ASA (Angle - Side - Angle) etc. If we can show that the relevant sides and angles of the two shapes (presumably triangles) are equal according to one of these criteria, we can prove they are congruent.