The question asks for the minimum additional information required to establish congruence between the two triangles using the AAS (Angle-Angle-Side) congruence criterion.

Given:

1. Both triangles already have one pair of angles congruent: $\angle ACB \cong \angle ACD$ (as shown with the green marks in the diagram).
2. A pair of sides ($\overline{AC}$) is common to both triangles.

AAS Criterion:

To prove triangles are congruent using AAS, we need:

1. Two pairs of corresponding angles congruent.
2. A corresponding side not included between those angles.

Analysis of Options:

• Option a ($\angle BAC \cong \angle B$): If $\angle BAC$ and $\angle B$ are congruent, this provides the second pair of congruent angles. Along with the side $\overline{AC}$, AAS is satisfied.

• Option b ($\angle B \cong \angle D$): While this suggests additional congruence between angles, it doesn't connect directly to the required setup for AAS for these triangles.

• Option c ($\angle ACB \cong \angle ACD$): This angle congruence is already given in the diagram, so it doesn't provide any new information.

• Option d ($\angle BAC \cong \angle DAC$): This isn't relevant for proving AAS directly since it doesn't help establish the required correspondence of two pairs of angles and one side.

Correct Answer:

The correct answer is a ($\angle BAC \cong \angle B$).

Would you like a deeper explanation or further help?

Here are 5 related questions to explore:

1. What is the difference between AAS and ASA congruence criteria?
2. Why do we need two angles and a side for AAS, instead of just one angle?
3. How does the shared side ($\overline{AC}$) help in this problem?
4. Can you prove these triangles are congruent using any other criteria, such as SAS?
5. How does knowing congruent triangles help solve real-world problems?

Tip:

Always check the diagram carefully to identify given information before analyzing the choices in a geometry question.