I would like to break my engagement to my cousin because she persists on neglecting the Sharee‘ah-approved dressing manner. She wears clothes that are disapproved by Sharee‘ah. I discussed this matter with her repeatedly but I no longer have hope that she will respond.

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His Slave and Messenger.

As far as we understand from the question, you proposed to a woman who was not adherent to Hijab, and when you tried to convince her to wear it, she wore a Sharee‘ah-disapproved form of Hijab. If it is the case, you were wrong when you got engaged to a woman who was neglecting the Sharee‘ah-approved Hijab and forgot the Hadeeth in which the Messenger of Allaah, sallallaahu ‘alayhi wa sallam, says: "A woman is married for [(either one of] these four (characteristics): her wealth, her lineage, her beauty and her religiousness. Choose a religious woman lest you lose." [Al-Bukhari and Muslim]

Religiousness should be the first criterion when a Muslim chooses a wife. A woman who neglects Hijab and persists on doing so has weak faith to a large extent. Therefore, she should not be chosen as a wife. Thus, if that girl persists on neglecting the Sharee‘ah-approved Hijab, it is permissible to break your engagement to her. You should seek a religious woman who has good morals.

Allaah Knows best.