Question:

The fast of the 9th of Av, the day on which our exile began, is always the same day of the week as the first day of Passover.  Is there any special significance to this?

Answer:

The ultimate purpose of the destruction of the 9th of Av is to reach the greater elevation of the final Redemption.

According to the Lubavitcher Rebbe, the fact that Tisha B'Av and Pesach fall on the same day of the week has significance.  In the Rebbe's words:

"The redemption of Pesach is the cure that preceded the illness of the 9th of Av.  Meaning, that on Pesach, the Jews were taken out of Egypt, and their state of enslavement and exile was nullified.  As the Maharal of Prague explains, when G-d took the Jews out of Egypt, he declared:  "They are my slaves" - and not slaves to slaves.

"In other words, even though it seems that on the 9th of Av, G-d abandoned His people (G-d forbid), the truth is that the 9th of Av is a continuation of the holiday of Pesach.  The ultimate purpose of the destruction of the 9th of Av is to reach the greater elevation of the final Redemption, as the prophet states:  'Like in the days of your exodus from Egypt, I will show you wonders.'"

Regarding the 9th of Av, our sages say:  "A lion [Nebuchadnezzar] arose in the month of the lion [Leo] and destroyed Ariel [the Holy Temple], in order that a lion [Moshiach] should come in the month of the lion and rebuild Ariel."

The Rebbe adds that this concept is alluded to in the name of the month -- Menachem Av.  "Beginning from the first day of the month of Av - even before the 9th of Av - we emphasize that the theme of the month is Menachem Av - our Father shall comfort us."  In fact in some sources, the month of Av is referred to simply as Menachem.

It is also important to note that our sages say that Moshiach's name is Menachem - since he will comfort us and lead us out of exile, immediately.