Engagement rhymes with:
disengagement, disengagement, disengagement, abridgement, acknowledgement, acknowledgment, arrangement, discouragement, disengagement, disgorgement, encouragement, enlargement, estrangement, infringement, judgement, judgment, management, mangement, micromanagement, misjudgment, mismanagement, nonmanagement, prejudgment, rearrangement, telemanagement
Engagement sounds like:
enchant, enchanted, engined, enhancement, enjoined, enjoyment