The question is asking which additional information is required to know that the two triangles in the image are congruent using the SAS (Side-Angle-Side) criterion.

SAS Criterion:

For two triangles to be congruent by SAS, two sides and the included angle between them must be congruent.

Analysis of the Image:

• Triangles in the Image:

  • $\triangle ZYX$
  • $\triangle WVX$

• Given Options:

  1. $\angle Z \equiv \angle V$
  2. $\angle Y \equiv \angle W$
  3. $\overline{ZY} \equiv \overline{VW}$
  4. $\overline{ZX} \equiv \overline{VX}$

Which Information is Needed?

To satisfy the SAS condition, we need two sides and the included angle to be congruent. Since the common side $\overline{ZX}$ is shared by both triangles, the key would be having:

1. Another pair of congruent sides.
2. The angle between these sides congruent.

The correct option should match a second side being congruent along with the angle between the two congruent sides.

Correct Answer:

The correct option would be Option 4: $\overline{ZX} \equiv \overline{VX}$, since you would then have two congruent sides ($\overline{ZY}$ and $\overline{VW}$, along with $\overline{ZX}$ and $\overline{VX}$) and the included angle ($\angle ZYX \equiv \angle WVX$).

Would you like more details, or do you have any questions? Here are some related questions you might consider:

1. How does the SAS criterion differ from other congruence criteria?
2. What would be needed to prove congruence using the ASA (Angle-Side-Angle) criterion?
3. How can you identify the included angle between two sides?
4. What are some real-life applications of triangle congruence?
5. How does knowing the congruence of triangles help in solving geometry problems?

Tip: Always ensure that the angle mentioned in the SAS criterion is the one formed between the two sides you are considering.