Their = belonging to them. There = place / “there is”. They’re = they are.
their
possessive: belonging to them
Test: Is something owned by “them”?
there
a place, or the start of “there is / there are”
Test: Can you swap in “here”?
they’re
contraction of “they are”
Test: Read it as “they are”.
“There” contains “here” (both places); “their” contains “heir” (someone who owns); “they’re” has the apostrophe for the missing “a”.
Five sentences. Apply the test above to each one before you pick.
_____ going to be late.
Leave the box over _____.
The students handed in _____ essays.
_____ is a problem with the printer.
I like them because _____ honest.
“They’re putting their bags over there.” If you can say which word does which job in that sentence, you have the rule.
The standalone possessive is “theirs”, with no apostrophe: “the choice is theirs”.
The grammar checker flags common mix-ups like this one by looking at the words around them, and the sentence checker can explain any sentence you’re unsure about.
“There is”. “Their” only ever means “belonging to them”.
Yes. Singular “they/their” for a person whose gender is unknown or who uses they/them is accepted by major style guides, including AP, Chicago and APA.