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Why do Pesach (פסח) and Tisha b'Av fall on the same day of the week, and why do we observe mourning customs during Pesach? The shiur develops the idea that Pesach contains potential for either redemption or destruction. If we had accepted Eretz Yisrael, the plagues would have turned permanently against the nations; our rejection on the ninth of Av meant those same destructive forces could target us.
Pesach (פסח) is time of potential destruction. Its "color" is techeiles, which relates to Makkas Bechoros. Idea is that purpose of Yetsias Mitzrayim was so we'd reach level of EY. If we'd be on level, sh'foch chamascha would be against goyim. If we'd reject EY - as we did on 9 Av, Makkos would turn against us. Thus, Pesach and 9 Av fall out on same day, and we have minhagei availus on Pesach. D'Rabanan's apply when Chazal saw dor would reach shlaimus in d'oraisa's only through this. Minhagim are same idea, but for single location.
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May a man who has already fulfilled his shofar obligation blow shofar for a woman on Rosh Hashanah? The Baal Itur forbids it while Rav Yehuda permits it. The shiur develops that the core dispute is not about violating shvus for another's benefit, but whether a man's blowing creates a "tekiah shel mitzvah" that enables the woman to be yotzei.
Why does the Gemara connect Purim to Tisha B'Av using "just like"—here you *decrease* joy, here you *increase* it? The shiur develops that Purim fixes the sin of Achashverosh's feast, where Klal Yisrael's desire to socialize with goyim revealed a physical disconnection from each other. Mishloach Manot and matanos la'evyonim create the physical unity—eating and drinking together—that is the tikkun for sinas chinam and the foundation for Kabbalas HaTorah.
