If m∠C = m∠Z and AC = XZ, then which of the following conditions is necessary for ∆ABC and ∆XYZ to be congruent?

The correct answer is BC = YZ.

To determine which additional condition makes △ABC ≅ △XYZ, let's carefully organize what we already know and apply triangle congruence theorems.

Given Information:
- m∠C = m∠Z (a pair of equal angles)
- AC = XZ (a pair of equal sides)

We need to find one more condition from the options that guarantees the two triangles are congruent.

Step 1: Identify the correspondence of vertices.

The triangles are named △ABC and △XYZ, so the vertex correspondence is:
A ↔ X, B ↔ Y, C ↔ Z

This means:
- Side AC corresponds to side XZ
- Side BC corresponds to side YZ
- Side AB corresponds to side XY
- ∠A corresponds to ∠X
- ∠B corresponds to ∠Y
- ∠C corresponds to ∠Z

Step 2: Map the known information onto the triangle.

We know:
- ∠C = ∠Z → this is the angle at vertex C in △ABC and vertex Z in △XYZ
- AC = XZ → this is the side between vertices A,C and X,Z respectively

So in each triangle, the angle is at one end of the known side. Specifically, the known side AC has the known angle ∠C at one of its endpoints. Similarly, XZ has the known angle ∠Z at one of its endpoints.

Step 3: Determine which congruence theorem applies.

We currently have:

- One angle: ∠C = ∠Z
- One side: AC = XZ (this side is adjacent to the known angle ∠C and ∠Z respectively)

To prove congruence, we need a second piece of information. Let's examine which theorem works:

▶ SAS (Side-Angle-Side): Two sides and the included angle between them must be equal.
If we add BC = YZ, then in each triangle we have two sides (AC and BC in △ABC; XZ and YZ in △XYZ) with the included angle between them (∠C and ∠Z). This perfectly satisfies the SAS congruence criterion.

The setup would be:
- Side AC = Side XZ ✓
- Included ∠C = Included ∠Z ✓
- Side BC = Side YZ ✓ (this is the new condition)

Step 4: Check why the other options do NOT work.

❌ AB = AC — This compares two sides within the same triangle (△ABC), not a side between the two triangles. It gives no new information about △XYZ and cannot establish congruence.

❌ AB = XY — AB is the side opposite to ∠C, and XY is the side opposite to ∠Z. Knowing AC = XZ and ∠C = ∠Z with AB = XY gives us a configuration that corresponds to SSA (Side-Side-Angle), which is generally not a valid congruence theorem and does not guarantee congruence.

❌ BC = AB — Just like the first option, this compares two sides within the same triangle (△ABC). It tells us nothing about △XYZ and is irrelevant to proving congruence between the two triangles.

Conclusion:

Adding the condition BC = YZ gives us two sides and their included angle equal in both triangles, satisfying the SAS (Side-Angle-Side) Congruence Theorem. Therefore, △ABC ≅ △XYZ.