Yes.
Both.
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Girls in Athens were just taught to do house work the were not aloud to be as productive as the boys they were to learn sewing and cooking mostly from there mothers though. if this does not help then let me know. Girls were also taught reading, writing, arithmetic , drawing, and skills you'll need to run a household.
Sparta and Athens were not rivals in 480 BCE - that came more than 30 years afterwards when the threat of Persia subsided. Sparta had promised to help them against the Persian attack on Athens in 490 BCE, but its army had arrived on the battlefield at Marathon too late. They remained on good terms and were natural allies in the Persian invasion in 480-479 BCE, providing the two largest contingents to the southern Greek coalition, led by Sparta, which repelled the invasion. Even if Athens had not been friendly with Sparta, it had no choice because it was a prime target of the Persian invasion and needed all the help it could get. It was not a case of Athens joining force with Sparta, it was Athens seeking the help of Sparta and its allies. They remained close allies for another 20 years, Athens helping Sparta put down a rebellion in its territory. They fell out when Athens started interfering in the affairs of the Peloponnesian League cities (which Sparta led) and Athens ignored Sparta's appeal to back off, leading to an all-out war.
The Athens sent a contingent to help the Sparta put down a serf uprising. The Athenians started to side with the serfs, and the Spartans sent them home. Relations soured.
Pheidippides, runner from Athens to Sparta to ask for help later from battle of marathon to Athens to report a victory
No. It defeated a Persian expeditionary force. Athens and Sparta were then friends. Pheidippides ran to Sparta to summon them to help, but the Spartans didn't arrive in time before the battle was fought.
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They continued their longstanding friendly and supportive relationship. Then Athens sent a contingent to help Sparta put down an uprising by its serf population, however the Athenians started to sympathise with the serfs and Sparta sent them home. Relations collapsed and they became hostile to each other.
Yes.
Allies Sparta and Athens had a falling out when the Spartan helots revolted and the forces Athens sent to help Sparta began to side with the revolutionaries, and Sparta sent them home. Athens turned the Delian league it had led against the Persian Empire into an empire of its own, and began to interfere in the cities of the Peloponnesian league led by Athens resulting in open war.
Yes, they won. But they only won because they had Sparta's army help them.