What the Quran tells us
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Surah An-Nur (24:31)
"And tell the believing women to lower their gaze, to preserve their chastity, and to show of their finery only what appears, and that they draw down their veil on their breasts; and that they only show their finery to their husbands, or their fathers, or their husbands' fathers, or their sons, or their husbands' sons, or their brothers, or their brothers' sons, or their sisters' sons, or Muslim women, or to the slaves they own, or to the impotent male servants, or to the immature boys who know nothing about the hidden parts of women. And let them not strike with their feet so that we know what they are hiding from their adornments. And repent to Allah all of you, O believers, that you may reap success.”
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Surah Al-Ahzab (33:59)
"O Prophet! Tell your wives, your daughters, and the wives of the believers, to draw back their large veils over themselves: they will be recognized more quickly and will avoid being offended."
These verses emphasize the importance and obligation of the veil and modesty for believing women.