D. SSS

To determine which theorem of triangle congruence shows that \(\triangle TUV \cong \triangle WVU\), we need to examine the given diagram and the marked congruencies:

1. \(\overline { TU} \cong \overline { WV} \)

2. \(\overline { TV} \cong \overline { WU} \)

3. \(\overline { UV} \) is a common side for both triangles.

Thus, the triangles \(\triangle TUV\) and \(\triangle WVU\) have three pairs of congruent sides:

- \(\overline { TU} \cong \overline { WV} \)

- \(\overline { TV} \cong \overline { WU} \)

- \(\overline { UV} \) (common side)

Since all three sides of one triangle are congruent to all three sides of another triangle, the correct congruence theorem is SSS (Side-Side-Side).

Supplemental Knowledge

Triangle congruence theorems are fundamental tools in geometry to prove that two triangles are congruent. The primary theorems include:

1. SSS (Side-Side-Side) Theorem: Two triangles are congruent if all three sides of one triangle are congruent to all three sides of another triangle.

2. SAS (Side-Angle-Side) Theorem: Two triangles are congruent if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle.

3. ASA (Angle-Side-Angle) Theorem: Two triangles are congruent if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle.

4. AAS (Angle-Angle-Side) Theorem: Two triangles are congruent if two angles and a non-included side of one triangle are congruent to two angles and a corresponding non-included side of another triangle.

Mastering triangle congruence theorems can significantly enhance your understanding of geometry. If you’re eager to dive deeper into these concepts or need help with other mathematical problems, UpStudy is here to guide you!