Which word means to examine or consider thoughtfully?

The word that means to examine or consider thoughtfully is "peruse." When someone peruses a text or material, they do not merely skim through it; instead, they engage with it more deeply, often paying attention to details and nuances. This term implies a careful, thorough reading or examination, which aligns perfectly with the idea of considering something thoughtfully.

The other choices do not convey this meaning. "Perplexed" refers to a state of confusion or being puzzled, which is unrelated to the act of examining. "Placate" means to calm someone down or appease them, and does not involve consideration or examination. "Placid" describes a calm or tranquil state, often used to contrast with disturbances, and does not pertain to the act of thoughtful examination at all. Thus, "peruse" is the appropriate choice for the definition provided.