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The question asks which relationship proves that triangles \( \triangle AEC \) and \( \triangle BDC \) are congruent.

Let's analyze the diagram:

- Triangles \( \triangle AEC \) and \( \triangle BDC \) are two overlapping triangles.
- \( \overline{AC} \) and \( \overline{BC} \) are equal, as indicated by the tick marks on both sides.
- Angles \( \angle EAC \) and \( \angle DBC \) are vertically opposite angles and therefore equal.
- \( \overline{AE} \) and \( \overline{BD} \) are also equal, as shown by the tick marks.

We now observe that:
- Two sides are equal: \( \overline{AE} = \overline{BD} \) and \( \overline{AC} = \overline{BC} \).
- One pair of angles is equal (vertically opposite angles).

This fits the **SAA** (Side-Angle-Angle) criterion for triangle congruence, as two angles and the non-included side are congruent.

Thus, the correct answer is **SAA**.

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Would you like further explanation or details on this? Here are some related questions to think about:

1. How do you distinguish between SSS and SAS congruence criteria?
2. What is the significance of vertically opposite angles in proving congruence?
3. Can triangles be congruent if they only share one equal angle?
4. What role do tick marks play in determining congruent sides in geometry?
5. How does the SSA condition differ from SAA?

**Tip:** In geometry, identifying shared angles and equal sides using given markers is key to solving congruence problems efficiently.

Which of the following relationships proves why ∆AEC and ∆BDC are congruent?

Understand how to prove that triangles AEC and BDC are congruent using the Side-Angle-Angle (SAA) criterion. Learn about congruent triangles, vertically opposite angles, and key geometry concepts, suitable for Grades 8-10 students.

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