The phrase "engage the attention" can be described using various synonyms that convey similar meanings. Some of these words include captivate, fascinate, allure, mesmerize, entrance, enthrall, engross, allure, and charm. These words imply that a person is highly absorbed or interested in something, like a particular topic or activity. Engaging someone's attention can be done through a wide range of means, from using stimulating visuals to telling evocative stories. It is important to engage the attention of your audience when you want them to listen closely to what you have to say or be willing to learn or consider new ideas.