Learn Italian: Ha alzato la mano per rispondere.

In Italian, you use the definite article (like la) instead of a possessive adjective for body parts and clothing when the owner is obvious. Since the subject of the sentence is doing the action, it is already clear whose hand is being raised.

Why does this sentence use avere (ha alzato) instead of essere?

Here, alzare (to raise) acts as a normal transitive verb taking a direct object (la mano), so it uses avere. You might be thinking of the reflexive verb alzarsi (to stand up or get up), which takes essere in compound tenses like the passato prossimo.

Why use per before the infinitive rispondere?

When expressing the purpose or goal of an action, you use per followed by an infinitive. It translates perfectly to "in order to" (he raised his hand in order to answer).

Mano ends in -o, so why does it take the feminine article la?

Mano is a common exception in Italian. It ends in -o but is feminine because it retains its gender from Latin. Keep in mind that its plural changes the ending to -i, but it keeps the feminine article: le mani.