Actually, you could meet both sentences meaning quite the same thing:
"Lequel est ton chapeau?": more common, popular, it implies more the idea of "which one among those hats (we're seeing/talking about, etc.)"
"Quel est ton chapeau?": a bit more formal, and more vague also. With this example, I can imagine that being asked at a cloakroom, there are plenty of hats and the guy lost his number or whatever.
It is clearer with another example: "What's your favourite animal?" or "Which is your favourite animal":
"Quel est ton animal préféré?": your favourite animal among all the existing (or extinct) animals in the universe, i.e. "What is...?"
"Lequel est ton animal préféré?": which one among those (here / we're looking at / we're talking about / etc.), i.e. "Which is...?"
So, non, it's not simply a question of "quel + noun" and "lequel" standing alone. Yes indeed, "lequel" will always be alone (or "lequel de ces animaux...", "which of those animals..."), but "quel" is not automatically seen just next to the respective noun, cf "Quel est ton animal préféré?", "Quelle est votre adresse", "Quel est le chemin le plus court?", etc.
Hope it helps.
Because even in English, that means something else:
"Is that your hat?" = you're showing one specific hat and you're asking IF that's mine
"Which is your hat?" = you have two or more hats in front of you and you're asking WHICH ONE is mine.
In French, those two different questions give respectively:
"Est-ce ton chapeau?" (quite formal). More informal: "Est-ce que c'est ton chapeau" or "C'est ton chapeau?". All 3 forms have exactly the same meaning.
"Lequel est ton chapeau?"
Basically, "quel" or "lequel" ask about "what (exactly)", "which", "which one", etc. Whereas a question like "Is that...?" (simple inversion) asks "whether or not" - that would be an inversion in French too (or "Est-ce que...?", or just a clear intonation showing it's a question).
"ton chapeau": "ton" is singular for "your"; "chapeau" is the singular form of "hat". "Chapeau" is a masculine noun in French (UN chapeau, LE chapeau), hence "ton" and not "ta".
plural "hats" would be "chapeaux", which differs only in writing and indeed sounds exactly the same as the singular form. Yet, with the rest of the question having "est" and "ton" (singular forms of "to be" and "your"), it can only be about "chapeau" in the singular.
therefore, the question word at the beginning of the question ("Which...?") MUST be singular AND masculine, in agreement with the main subject of the sentence, i.e. "ton chapeau". That's a basic grammatical rule in French: articles, pronouns, adjectives, roughly every word grammatically related to a noun, change their form according to the gender and number of the noun they relate to.
so, here: "lequel" = singular masculine form for the equivalent of "Which one" in English. While "lesquelle", as you wrote it, doesn't even exist. If not "lequel", It can only be either "lesquels" (masc. plural), "laquelle" (feminine singular) or "lesquelles" (feminine plural), so "lesquelle" is nonsensical since it has the mark of plural at the start ("les-") but no at the end ("-quelle").
But again here, with the rest of the sentence, you should deduct that it is "lequel".
- so even though "quel", "quels", "quelle" and "quelles" all sound the same, and even if you can't hear the difference yet between the "le" and "les" sounds, however you should know those differences and their reasons why - grammatically and so when you write.
Okay, this is really bizarre. French must've changed in the 30 years that I had to speak/write it, or maybe somewhere I just wasn't taught this whole "laquelle" and "lequel" thing. Seriously, this is messing me badly on these test-outs. We were always taught "que", "quel", and "qui" and the various. And I don't recall (from my time living in France, but that was long, long ago) any "laquelle"s or "lequel"s. Am I just...going senile or something?
lequel, laquelle, lesquels, lesquelles are interrogative or relative pronouns.
They generally correspond to "which one, which ones" or "which/what" depending on the construction of the sentence:
- among these hats, which one is yours? = parmi ces chapeaux, lequel est le tien / lequel est à toi ?
First, French does not have that distinction between "what" and "which".
As for English, simply put, you ask "what [something]" when the choice, the possibilities of answers are very wide:
What is your favourite animal [among all the animals on Earth] ? = Quel est ton animal préféré [parmi tous les animaux de la Terre] ?
Of all animals, what is your favourite ? = De tous les animaux, lequel est ton préféré ?
BUT, if you talk about specific animals (whether in a limited context or a certain category), then you should ask "which":
Which kitten is your favourite (out of the kittens you have at home) ? = Quel chaton est ton préféré / Quel est ton chaton préféré (parmi les chatons que tu as à la maison) ?
Your kittens are so cute, which [one] is your favourite ? = Tes chatons sont si mignons, lequel est ton préféré ?
As you can see, there is no difference in French (each time, we used "quel" or "lequel") whereas English uses "what" for general questions and "which" for specific, limited choices.
Take the famous "What's your number ?" or "What's your name ?" (Quel est ton numéro / ton nom ?): there is an ocean of possibilities and probabilities which to answer from.
So here, in this exercise, asking "What hat is yours?" would sound weird, because it would be like wanting to know what hat is potentially mine out of all the hats in the world (does not make much sense...). The context here could be, for instance, a cloakroom and the attendant asks the visitor which hat is theirs among the hats in the room... Limited choice, then "which".
Google "what vs which". You should get a lot of information. I'm a native English speaker, but even I don't know how to explain the difference. Although they are similar and can sometimes be used interchangeably, there are some situations wherein using one and not the other will sound strange and bad.
"Lequel" is a pronoun, not an adjective : it must be used alone, not with the noun it refers to. Otherwise, you use "quel" : "Quel chapeau ... ?". e.g. "Quel chapeau choisir ?" = "Which hat to choose ?". If we know what we're talking about (hats) and we just want to imply the noun, we then say "Lequel choisir ?" = "Which one to choose ?"
"Ton" is a possessive determiner ; determiners act like adjectives, i.e. they are used with the noun they refer to. If you want to imply the noun, you must use the equivalent pronoun; with "ton" (your), it is "le tien" (yours). E.g. "Mon chapeau est bleu, le tien est rouge" = "My hat is blue, yours is red" (if you swap persons, it gives "Ton chapeau est rouge, le mien est bleu").
So your sentence should be either the given translation in the exercise, or "Quel chapeau est le tien ?"
In this lesson, you are learning interrogative pronouns.
"Lequel, laquelle, lesquels, lesquelles" are the interrogative pronouns used in the French sentences, and you have to translate them to pronouns, not to adjectives.
Which hat is yours? back translates to "quel chapeau est à toi ?", with "which/quel" modifying "hat/chapeau", as adjectives.
"Lequel est ton chapeau ?" should therefore best translate to "which one is your hat?".
Not only are you learning interrogative pronouns ("lequel" vs the adjective "quel"), but you are learning the difference in meaning between them.
Which one and which are not interchangeable either, and "lequel/which one" imply a narrower universe of choice (2 to 3 hats) than "quel/which" (more than 3 hats).