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It is important to note that Archaeopteryx is not the first transition between reptile and bird (see transition from dinosaur to bird). Still Archaeopteryx is the best known, having been discovered only a few years after the publication of Origin of the Species and subject to much controversy with suspicions of it being a hoax (accusations, which so far, have not stood up well).
In 1985, some non-creationists (but also non-experts) argued that the fossil was actually a hoax, with feather impressions having been added to an existing reptile fossil. One of the accusers was Fred Hoyle, an astronomer whose unfortunate misunderstandings of biology gave us Hoyle's fallacy and the directed panspermia hypothesis with which ID makes much hay. As tempting as it might be for them to once again argue from Hoyle's authority and call Archaeopteryx "another Piltdown man", only a few creationists do. (Among many difficulties with such a suggestion is that the discoverer was Richard Owen, a noted opponent of Darwin, though not entirely against evolution.)
Instead, creationists like to call it "fully bird" and downplay the most noticeable reptilian features: claws and teeth. They say that ostriches have larger claws than Archaeopteryx in their wings, and believe that since many reptiles (such as some snakes) do not have teeth, it makes the fact Archaeopteryx has teeth entirely pointless.
These arguments rely on a form of species essentialism as applied to classes such as "bird" and "reptile". Both of these are groups that can be well-defined in a broad sense but naturally involve small exceptions as a result of some facet being dropped (as with loss of teeth) or retained (as with presence of claws). Loss of a trait is actually reasonably understood by creationists, since they can just dismiss it as "devolution". Retention of an ancestral trait is harder to grasp (see vestigial structure).
Nature is not required to limit itself to our nice neat categories, and evolution even predicts a blending of these categories in the form of transitionals like archaeopteryx. Even if we had no fossil record, the ostrich's claws would still serve as an excellent piece of evidence for their descent from reptiles, not as some kind of cosmic sign that the Platonic Ideal of birds is permitted to include wing claws.
(In addition, the possibility of traits persisting or convergently re-appearing along a particular branch of the tree means this argument can be repurposed to dispute any transitional. For example, the rare spotted handfish uses its limbs to literally walk on the bottom of its shallow habitat; from this, a creationist could say "Walking is a fish trait, so all fishapods are fully fish," without considering that a few steps from this argument and we may as well consider amphibians "fully fish" as well, as cladists do.)
Pointing to modern bird species such as the ostrich and hoatzin (which loses claws in youth) is just a form of presentism. Hypothetically, they could have gone extinct long ago, in which case such features would be confined to fossil birds and no living ones. Would creationists still be happy to use them as justification for the birdiness of wing claws? Conversely, Archaeopteryx could have survived to the present (in fact, creationists do think it was contemporaneous with humans). Suppose a live Archaeopteryx had been discovered in a world where no living or fossil bird had been found with teeth or wing claws, other than in embryonic development. On what grounds would creationists call it "fully bird"?
Answers in Genesis like to claim that Archaeopteryx was just an ordinary bird and not a transitional form, which is an example of creationists moving the goalposts of what a "transitional form" is to satisfy their own agenda.
A 2011 analysis found Archaeopteryx to have more in common with raptor dinosaurs than with birds, essentially making it "another feathered dinosaur" in retrospect. This was treated as some kind of blow to evolution by Uncommon Descent . Perhaps the creationist author assumes that evolution actually requires species essentialism. Whatever the reason, such derision poses a considerable problem: creationists had always been adamant that it was 100% bird, now and forevermore, while biologists had basically adjusted from 60% bird to 40% bird, insofar as such measurement is even meaningful. Archaeopteryx is hard to classify because the classification is inherently fuzzy. (It's almost certainly not ancestral to modern birds, which is something else creationists ignorantly made a big deal about.)
The increasing preponderance of feathered dinosaurs may lead to creationist difficulties analogous to their troubles in agreeing on which hominids are fully ape and which ones fully human, with Archaeopteryx being the locus of the most disagreement due to the combination of its iconic "fully bird" status and the preponderance of evidence that it was even more reptilian than many other therapods. We can only hope.
Recent studies on fossil feathers of Archaeopteryx have revealed the coloration of the feathers themselves, showing them to be shades of black and white. This is seen in many modern birds such as magpies and penguins. It may have helped in harnessing shade for various reasons. 
- Teeth present in the adult stage.
- Dinosaur-shaped pelvis.
- Long tail.
- Not all fingers fused.
- Neck attaches to skull from the rear instead of from below.
- Metatarsals not fused.
- Gastralia (ventral ribs) present.
- Trunk vertebrae not fused.
- See the Wikipedia article on Archaeopteryx.
- Answers in Genesis
- "Archaeopteryx" on AnswersinGenesis.org
- "Archaeopteryx is NOT a hoax—it is a true bird, not a “missing link" on AnswersinGenesis.org
- "Remember the Icon of the First Bird, Archaeopteryx? Word is, it’s not a bird" on UncommonDescent.com