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Hello, I am a math teacher and I need to write proofs with line names, which is AB with a two-sided arrow on top. I can't find a way to do it. There are instructions how to write AB with an arrow on top, but that it a vector or a ray, but not a line. Thank you!

Format > Character > Font effects adds a line.
Look at the Math package of AOO or LO. File > New > Formula.

Windows allows you to write mathematical formulae with Start > Accessories > Math Input Panel.
Google for maths fonts. Latex is excellent for math formulae. There are on-line sites for math formulae like Online Mathematics Formula Editor

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(Off topic)

I'm also a math teacher (Germany), now retired for more than a decade, but I cannot remember to often have read mathematical texts containing the notation for straight (unlimited) lines you described. Starting with texts (transcribed) from ancient Greece lines defined by two points were denoted by simply putting the two one-letter-point-names side by side. As an example "AB x BC = B if A, B, C not are collinear" should tell the obvious. (When I wrote a paper where I wanted to denote the operation of joining two given points A and B by a straight line g explicitly I used "A..B = g".)

Where and since when is the notation described in the question common or even mandatory?

@Lupp Different notations are used in Europe and America. In the USA, the line is denoted by a double-sided arrow above, and a line segment by a line above. Since I write this for school aged kids, I need to keep the notations as they are used in the textbooks here.

John_Ha wrote:Latex is excellent for math formulae.

... and the add-on amsmath supports double-headed arrows. See Typesetting arrows above a pair of letters to indicate a line
The brute force and ignorance AOO method is to use the Draw component of Writer to add the arrows or create an image of arrows and add it above the letters.