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Blessings over the Torah
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Shavuos is the Holiday during which we commemorate, primarily, Hashem's giving us the Torah -- both the Written and Oral Torah -- at Mount Sinai, over 3,315 years ago.
The Laws of the Holiday of Shavuos are actually taught in the Laws of Pesach (Passover), because in a sense Shavuos is a continuation of Pesach. Hashem took us out of Egypt in order to give us the Torah, so that we could, via the Torah, become a kingdom of priests, a holy nation.
It is the Custom in most communities to decorate the synagogues and homes for Shavuos with flowers and branches. It must be done before Shavuos, because on Shavuos it is forbidden to cut any growing vegetation. This is done to remember the joy of the Giving of the Torah, because, among other reasons, at that time Hashem performed a miracle and caused sweet-smelling plants to grow at Mount Sinai, where nothing usually grows.
However, it is forbidden to use an entire tree, whether large or small, because that is a gentile custom done on one of their celebrations. The Torah forbids us to do the same customs as gentiles.
It is Customary for a Jewish person to immerse in a mikvah the day before Shavuos.
On what would be the fiftieth night and day of Sefirah (though we actually count only forty-nine nights), we keep the Holiday of Shavuos. The prayers are pretty much the same as any of the three cyclic Holidays (Pesach, Shavuos, Sukkos), but at the appropriate times in the prayers we say "the day of the Holiday of Shavuos, the time of the Granting of our Torah...."
And like the other Yomim Tovim (Jewish Holidays), outside of Israel we hold two days of Yom Tov (Holiday).
We wait and pray later than usual on the night of Shavuos, because the Torah says that we must count seven full weeks of Sefiras Ha'Omer, starting from the second night of Passover until the night before Shavuos (Leviticus 23:15).
The Torah tells us,
"You shall count seven complete weeks after the day following the [Passover] holiday when you brought the omer as a wave offering, until the day after the seventh week, when there will be [a total of] 50 days. [On that 50th day] you may present new grain as a meal offering to God.
-- Leviticus 23:15-16
That fiftieth day is the Holiday of Shavuos.
The Torah specifically says "seven complete weeks." Therefore, we may not pray Maariv (the nighttime Prayers) at the beginning of Shavuos until the entire period of "seven complete weeks" has completely and definitely ended.
Usually, when Holidays begin, and when Shabbos begins, it is generally permitted to begin the Holiday early, and indeed, sometimes it is preferable. But we may not begin the Holiday of Shavuos early, so that the Count of Omer should be a complete seven weeks. Therefore, both nights of Shavuos we wait until the stars come out, and do not begin praying until then.
It is the Custom to remain awake all night the first night of Shavuos and learn Torah until dawn. This is because of something we are taught in a Medrash:
The night before the giving of the Torah, that is, the night of the very first Shavuos, all of the Children of Israel went to sleep. When morning came, they overslept. To amend this, it is the Custom to stay awake the night of Shavuos until dawn, studying Torah.
Once the dawn arrives, it becomes forbidden to study Torah without first saying the Blessings over the Torah again. The new dawn comes with a new requirement to say the Morning Blessings. However, we may not say many of those, because we have been awake all night, and we have not slept.
Therefore, what many people do is go to sleep, wake up a few hours later to pray the Morning Prayers, and at that time say all the Morning Blessings.
It is imperative that you wake up in time to pray all the prayers before the final time allotted for the Morning Prayers. Just as the Torah mandates set times for the offering of the Holy Temple Sacrifices, and they may not be brought once that time has passed, so too, the prayers, whose times are based and set according to the times of the Holy Temple Sacrifices, may not be recited (by men) after their time has passed. Once the time of the Morning Prayers has passed without your having said them, that is a loss that can never be recovered. (Women, however, are not bound by time the same way that men are. The rules of time-related Laws are different for them, in that in some Laws they follow time like men do, but in most things they follow Laws that are associated with their own bodies and bodies' cycles.)
Many people, however, are not certain they'll be able to get up again so quickly, after being awake all night. So they use another solution.
They find someone who has slept at least thirty-five minutes, and they have that person say the Blessings for them. The Halachah (Jewish Law) is that if someone who is also required to say those Blessings, says them out loud for you, and you answer Amain, it is considered as if you said those Blessings. However, this works only if both the person saying the Blessing and the person hearing the Blessing intends to fulfill the obligation of both people.
These people then begin the Morning Prayers immediately, before going to sleep. They begin to pray at dawn, which actually is the best time to pray (though the Laws of this are complicated, so don't do it unless you are with a minyan (quorum group for prayers) that knows how to do it right). If you do that, you may not say the brachos (blessings) over washing your hands and Asher Yatzar ("Who has formed mankind...") unless you first use the restroom (Mishnah Brurah 494:1).
The Laws of the other brachos are also complicated, and many people prefer to hear them said by someone who has slept, and answer amain and thus fulfill their obligation.
This includes Birchos HaTorah, the Blessings before Studying Torah that we must say each morning. After hearing Birchos HaTorah, learn the Torah that we learn every day after those brachos, which includes the Priestly Blessing and so forth (Mishnah Brurah 47:28).
However, the best way to solve the question of Birchos HaTorah is to sleep at least an hour or so during the day BEFORE Shavuos, for the purpose of being able to stay awake that night. If you do that, then you may say Birchos HaTorah by yourself, without any doubt (Mishnah Brurah, ibid, citing Rabbi Akiva Eiger).
However, this does not apply to two brachos: E-lokai Nishamah, which refers to the soul returning to the body in the morning when you wake up; and the brachah (blessing) of "Who removes sleep from my eyes and slumber from my eyelids..." If you have not woken up recently from an hour's (or more) sleep, then those two brachos you have to hear from someone else.
Others go to sleep at dawn and wake up to pray at the normal hour that they start the prayers every Yom Tov. (Perhaps they set an alarm clock.)
If you combine that with sleeping during the day before Shavuos, then you may say all the Morning Brachos when you wake up to pray.
Many Chassidim have the Custom to go to the Mikvah at dawn on Shavuos morning.