There are many different synonyms for the phrase "accepted that offered". Some common alternatives include "took up the offer", "embraced the opportunity", "seized the chance", "agreed to the proposition", "grasped the possibility", and "accepted the invitation". Depending on the context in which the phrase is being used, other potential synonyms could include "welcomed the proposal", "acknowledged the offering", "recognized the value", or "responded affirmatively". Ultimately, the choice of synonym will depend on the tone and meaning you are trying to convey, as well as the specific vocabulary and phrasing that best suits your intended audience and purpose.