Meine Schwester sucht ihr Geodreieck, bevor die Matheprüfung beginnt.

Because meine Schwester is the subject of the main clause, so it is in the nominative case.

• meine Schwester = my sister as the person doing the action
• meiner Schwester would be dative or genitive in other contexts

In this sentence, Meine Schwester sucht ... means My sister is looking for ..., so nominative is required.

German main clauses usually follow the V2 rule: the finite verb goes in the second position.

Here the structure is:

• Meine Schwester = position 1
• sucht = position 2
• ihr Geodreieck = rest of the clause

So Meine Schwester sucht ihr Geodreieck is normal German word order.

Here ihr means her.

It is a possessive determiner, showing that the set square belongs to the sister.

Because Schwester is feminine, German uses:

• sein = his / its
• ihr = her

So:

• ihr Geodreieck = her set square

Because Geodreieck is a neuter noun: das Geodreieck.

The verb suchen takes a direct object, so the object is in the accusative case. For a neuter singular noun in the accusative, the possessive determiner has no extra ending here:

• nominative neuter: ihr Geodreieck
• accusative neuter: ihr Geodreieck

So the form stays the same.

You would see ihren with a masculine accusative noun, for example:

• Sie sucht ihren Stift. = She is looking for her pen.

A Geodreieck is a school geometry tool, often translated as set square or triangle ruler.

It is the triangular plastic tool used for:

• drawing lines
• measuring angles
• geometry work in school

It is a compound noun:

• Geo- = geometry-related
• Dreieck = triangle

Like all German nouns, it is capitalized: Geodreieck.

Because bevor introduces a subordinate clause, and German normally separates subordinate clauses with a comma.

So:

• Meine Schwester sucht ihr Geodreieck,
• bevor die Matheprüfung beginnt.

That comma is required in standard German.

Because bevor introduces a subordinate clause, and in German subordinate clauses the finite verb usually goes to the end.

So:

• bevor die Matheprüfung beginnt

not:

• bevor beginnt die Matheprüfung

This is one of the most important word-order patterns in German.

Because Matheprüfung is a feminine noun: die Matheprüfung.

In the subordinate clause, it is the subject of beginnt, so it is in the nominative case:

• die Matheprüfung beginnt = the math exam begins

Also, German often uses the definite article in places where English might or might not use one. Here it refers to a specific exam.

Yes. German very often forms compound nouns by writing them as one word.

So:

• Mathe = math
• Prüfung = exam / test
• Matheprüfung = math exam

This is completely normal in German. English often uses separate words where German prefers one compound noun.

Yes. German present tense often covers both:

• looks for
• is looking for

So Meine Schwester sucht ihr Geodreieck can naturally mean:

• My sister looks for her set square
• My sister is looking for her set square

In this context, English would usually say is looking for, but German simply uses the present tense sucht.

Bevor is a conjunction meaning before, and it introduces a clause.

• bevor die Matheprüfung beginnt = before the math exam begins

Vor is usually a preposition, followed by a noun phrase:

• vor der Matheprüfung = before the math exam

So:

• bevor
  • clause
• vor
  • noun

That is the key difference.

Yes. German allows that.

You could say:

• Bevor die Matheprüfung beginnt, sucht meine Schwester ihr Geodreieck.

That is also correct.

Notice what happens in the main clause after the subordinate clause:

• the verb sucht still comes first in the main clause
• meine Schwester follows it

This is because the whole bevor clause takes the first position, so the finite verb of the main clause must come next.

Yes. It is a complete clause:

• die Matheprüfung = subject
• beginnt = verb

It means the math exam begins.

Inside the full sentence, it becomes a subordinate clause because it is introduced by bevor:

• bevor die Matheprüfung beginnt = before the math exam begins

So it is grammatically complete, but dependent on the main clause because of bevor.