Khula

Khula (Arabic: خلع‎‎) is a procedure through which a woman can divorce her husband in Islam, by returning the dower (mahr) that she received from her husband. Based on traditional fiqh, and referenced in the Quran and hadith, khula allows a woman to initiate a divorce through the mutual consent of the husband or a judicial decree.

2:228 "And divorced women should keep themselves in waiting for three menstrual cycles; and it is not lawful for them to hide what God has created in their wombs, if they believe in God and last day; and their husband have more right of taking them back during this, if they desire reconciliation; and for them are like from them in recognised manner; and men are a degree above them; and God is Mighty and Wise."

2:229 "Divorce is twice. Then, either keep [her] in an acceptable manner or release [her] with good treatment. And it is not lawful for you to take anything of what you have given them unless both fear that they will not be able to keep [within] the limits of Allah . But if you fear that they will not keep [within] the limits of Allah, then there is no blame upon either of them concerning that by which she ransoms herself. These are the limits of Allah , so do not transgress them. And whoever transgresses the limits of Allah - it is those who are the wrongdoers."

4:128 "And if a woman fears from her husband contempt or evasion, there is no sin upon them if they make terms of settlement between them - and settlement is best. And present in [human] souls is stinginess. But if you do good and fear Allah - then indeed Allah is ever Acquainted, with what you do, ."

65:4 "And those who have lost the hope of courses and those who do not have courses, if you doubt, their iddah is three months; and for those who are pregnant (whether they are divorced or their husbands are dead), their iddah (prescribed period) is until they deliver their burdens..

The most well known story that references khula and serves as the basis for legal interpretations is the story of Jamilah, the wife of Thabit ibn Qays:

Narrated Ibn 'Abbas: The wife of Thabit bin Qais came to the Prophet and said, "O Allah's Apostle! I do not blame Thabit for defects in his character or his religion, but I, being a Muslim, dislike to behave in un-Islamic manner if I remain with him." On that Allah's Apostle said to her, "Will you give back the garden which your husband has given you as Mahr?" She said, "Yes." Then the Prophet ordered to Thabit, "O Thabit! Accept your garden, and divorce her once."

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Wikipedia