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The reason, Allah knows best, that supplication is answered at this particular time is that as long as a person is waiting for the prayer, he is deemed to be in prayer—and prayer itself is a setting where petitions are granted, for therein the servant communes privately with his Lord.
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Shaykh al-Islam stated: “Supplication at the close of the prayer, before one exits it, is prescribed by the widely transmitted Sunnah and by the unanimous agreement of the Muslims. Virtually all of the invocations connected to the prayer were offered by the Prophet, peace and blessings be upon him, within the prayer itself, and he commanded that they be made there. This befits the condition of a worshipper who is facing his Lord and confiding in Him. He should therefore choose the supplication dearest to Allah, and let him pray with humility and proper decorum, for a petition issuing from a heedless heart will not be answered.”