Verb Conjugation -- Acheter

In Lesson 7.3 is the sentence, “Un cadeau acheté par un ami,” translated to, “A gift was bought by a friend.”
 
Why was the present tense used; shouldn’t it have been the passé composé and read, “Un cadeau a acheté par un ami”?
 
The action in the sentence is a purchase that happened at one point of time in the past (needing the passé composé).  The action wasn’t on-going and continuing into the present, nor is the action right at this moment (each needing the present).

I think it's supposed to be a clause, and not a complete sentence, just like the example underneath it, so there shouldn't be a "was" in the English translation:

Un cadeau acheté par un ami - a gift bought by a friend

Toursan, there’s something with “Un cadeau acheté par un ami” that neither of us at first saw.  Clearly, the English translation is a complete past tense sentence in the passive voice; however, if the French is intended to be a sentence, it has an error.  The verb “acheté” is only used in the conjugation of acheter when the auxilliary avoir is used, such as in one of the past tenses.
 
As you noted, the correct passé composé in the active voice is, “Un ami a acheté un cadeau,” but I think the passive voice should be, “Un cadeau a été acheter par un ami.”  The latter because after another verb, the second verb is in the infinitive.  “Un cadeau a été acheté par un ami” is incorrect because “acheté” is only used after an auxilliary.
 
“Une fenêtre cassée par un enfant” may not be as straight forward as it at first appears.  Here “cassée” is an adjective describing the window.  The third person present singular conjugation of “casser” is “casse”; in no tense of casser's conjugation does cassée appear.
 
Finally, although at first glance the two examples appear to be similar construction, in reality they aren’t.

Hi Diane, no actually, you would also use the past participle in indirect construction as I'd indicated: «Un cadeau a été acheté par un ami.».

The past participle (by itself) is actually quite versatile. There isn't this rigid stricture of using the p.p. just for forming simple passé composé (along with the rules of être for «les verbes de déplacement» and avoir for all else); it's also used for other constructions to indicate something past and functions as an adjective (like the "broken window" example, or a missed opportunity «une opportunité manquée»).

Unfortunately, I can't remember the specific grammar workbook that I was using at the time when there were exercise questions like these, but I do have Grammaire Progressive du Français (blue book) handy, and on Chapter 52 (Les Relations Logiques), you can see a similar construction used to explain the usage of comme and parce que:

Le match a été annulé parce qu'il pleuvait.
Comme il pleuvait, le match a été annulé.

And yes, because it is being used as an adjective, it has to agree in gender and number.

Ah, found the book! It's The Ultimate French Review and Practice (Stillman and Gordon), Chapter 27: The Passive Voice and Substitutes for the Passive.

Some examples:
La voiture a été réparée par Max, pas par moi. - The car was repaired by Max, not by me.
Les bicyclettes ont été vendues. - The bicycles were sold.

I see, Toursan.  This leads me to wonder just what the lesson's writer had in mind.  Certainly, the upper of the two examples has one meaning in French and a different meaning in English.
 
Rocket French often uses parts of speech and/or constructions not yet studied, and that’s okay as they don’t confuse the student.  Furthermore, the curious can easily look them up to further his learning.  In the Lesson 7.1 examples, however, the unwitting student could think that acheté and cassée are used as verbs, when in reality the former is used an adjective and the latter is an adjective.
 
The two examples could be written “Un cadeau bleu par un ami” and “Une fenêtre bleu par un enfant,” and be just as grammatically correct as are the lesson’s examples.  They don’t make sense, but that’s another matter.
 

Bonjour à vous deux!

\je suis vraiment désolée de ne pas avoir vu cette erreur. Let me try and clear this up. You are fully justified in think something looks a bit fishy here because actually the \english translation that has been given is indeed wrong. It should simply read:
"Un cadeau acheté par un ami" = "A gift bought by a friend"
As you have pointed out above, the verb "acheter" which appears here in the passé composé form is not taking the role of an active verb but rather an adjective and expresses that the gift was bought in more of a passive manner.

I am very sorry for the confusion and will get on to fixing it straight away.

   -   Marie-Claire