- Homologous structures (Option 1): These are structures that have a similar origin but may serve different functions in different species. The thorn of Bougainvillea and the tendril of Cucurbita are both modified stems and are therefore homologous structures.
- Analogous structures (Option 2): These are structures that serve similar functions but have a different evolutionary origin. This is not the case here.
- Vestigial structures (Option 3): These are structures that have lost their original function. Neither the thorn of Bougainvillea nor the tendril of Cucurbita fits this description.
- Developing structures (Option 4): This term is not typically used in evolutionary biology to describe structures. Step 2: Conclusion.
Both the thorn and tendril are homologous structures.
Final Answer:
\[
\boxed{\text{The correct answer is (1) Homologous structures.}}
\]