Given Information

• Triangle PQR and triangle STU
• Angles P and S each measure 30°
• QR = 16 and TU = 12
• Statement I: The length of PQ is 4/5 the length of ST
• Statement II: The length of PR is 4/3 the length of SU

Triangle Similarity Requirements

For two triangles to be similar, we need one of these conditions:

• AAA: All corresponding angles are equal
• SAS: Two pairs of corresponding sides are proportional with equal included angles
• SSS: All three pairs of corresponding sides are proportional

Analysis of the Problem

Let's examine the side ratios we can determine:

From the given information:

• QR/TU = 16/12 = 4/3

From Statement I:

• PQ/ST = 4/5

From Statement II:

• PR/SU = 4/3

The Critical Issue

For triangles to be similar, all corresponding sides must be proportional with the same ratio. However, we have:

• PQ/ST = 4/5 = 0.8
• PR/SU = 4/3 ≈ 1.33
• QR/TU = 4/3 ≈ 1.33

Since PQ/ST ≠ PR/SU ≠ QR/TU (specifically, 4/5 ≠ 4/3), the sides are not proportional with a consistent ratio.

Conclusion

Neither Statement I nor Statement II is sufficient because the fundamental requirement for triangle similarity—that all corresponding sides be proportional—cannot be satisfied with the given measurements. The ratios 4/5 and 4/3 are different, which means these triangles cannot be similar regardless of which additional information we consider.

The correct answer is D: Neither I nor II is sufficient.