In verbis “mater filiae librum dat” “filiae” dativus est, et in sententia “cum amica ambulo” nomen “amica” in ablativo est.

Because filiae is the person who receives the book. With a verb like dat from dare, Latin often has:

• a subject: mater
• a direct object: librum
• an indirect object / recipient: filiae

Latin marks that recipient with the dative case. In English, we often use to for this idea, as in the mother gives a book to the daughter, but Latin can show it just by the case ending.

That is a very good question, because filiae is an ambiguous form. It can be:

• genitive singular
• dative singular
• nominative plural
• vocative plural

So how do we know which one it is? We use the sentence structure.

In mater filiae librum dat, the verb dat naturally suggests a giver, a thing given, and a recipient. Since librum is the thing being given, filiae is best understood as the recipient, so it is dative singular.

In mater filiae librum dat:

• mater is nominative singular: the subject, the one doing the action
• librum is accusative singular: the direct object, the thing being given
• filiae is dative singular: the recipient

So the sentence shows a very common Latin pattern: nominative + dative + accusative + verb.

Because the preposition cum meaning with takes the ablative case. In this sentence, amica is the person accompanying the speaker, so Latin uses the ablative after cum.

So:

• cum = with
• amica = ablative after cum

This is a standard rule: cum as a preposition of accompaniment governs the ablative.

Because without macrons, the forms look identical.

For a first-declension noun:

• nominative singular: amica
• ablative singular: amicā

If macrons are not written, both appear as amica. So the spelling alone does not tell you the case. The preposition cum tells you that the form must be ablative.

As a preposition meaning with, yes: it takes the ablative.

Examples:

• cum amica
• cum puero
• cum magistrō

But there is also cum used as a conjunction, meaning things like when, since, or although. That is a different use. In that use, it does not govern a noun in the ablative; instead, it introduces a clause.

Because Latin often expresses grammatical relationships through case endings rather than separate words. English usually needs word order or prepositions such as to and with, but Latin can build that information into the noun form itself.

So:

• English: to the daughter
• Latin: filiae

The dative ending already tells you to the daughter or for the daughter, depending on context.

Those are just two slightly different ways of labeling a case.

• dativus est = is dative
• in ablativo est = is in the ablative

Both are normal ways to describe case. The difference is mostly stylistic. The grammar of filiae and amica is not different because of that wording.

Yes. Latin word order is more flexible than English word order because the cases show each word’s role.

For example, these can mean the same basic thing:

• mater filiae librum dat
• librum mater filiae dat
• filiae mater librum dat

The cases still tell you:

• mater = subject
• librum = direct object
• filiae = indirect object

Changing the order usually changes emphasis, not the core meaning.

No. They are different cases with different jobs.

• filiae is dative because it shows the recipient
• amica is ablative because it follows cum and shows accompaniment

So even if both nouns are connected to the verb, they are connected in different ways. The dative answers something like to whom?, while the ablative with cum answers with whom?