My answer is "other", because the polling question is poorly structured/worded, with all due respect to the OP.
For starters, there's no such thing as "transgendered". Transgender means their biological sex and gender identity are opposite of each other. There's no "ed" necessary.
The OP mentions nothing about their present hormone estrogen levels, or whether or not they had bottom surgery, or if they transitioned before/during puberty.
My belief here is that High School sports should be open to everybody. As long as their estrogen levels are at biological female levels, they should be permitted to join the girls team. High School sports are not really about winning. They are about education and participation. It's about developing friendships, and learning about hard work, discipline, and being a team player. A greater sin is depriving people of that education simply because they have a neurological condition.
For college, I think the current NCAA rules are fine and have experienced pretty much zero problems. They require the trans female to be on estrogen for at least 2 years straight, and their estrogen level must be at a certain level to play. I have no problem if they tighten or loosen the currently restrictions. It's up to them.
Without asking about the female's current hormone levels, the poll question makes less sense to me.
My answer is YES, they should be permitted to play on female teams as long as they get doctor approval and have taken hormones for a given time period.
The problem here is when we start banning transgender females from sports (which is a very small sample btw), then we will eventually start to go down a slippery slope. Does this mean Caster Semenya can't play in the female leagues? She was born with a vagina, but has XY chromosomes and produces high level of testosterone. Does this mean a 6'7 female basketball can't play basketball because she's too tall? Back when Jackie Robinson played baseball, many people didn't want him to play because he was black and black people were seen to have a genetic advantage in sports.
My other problem here is we only pick on the transgender females which are actually good, and train extremely hard. The ones which are not, get ignored.