Question: Are both Chow Chow and American Hairless Terrier examples of dog breeds?

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Rare breed (dog): Rare breed (dog) is any breed of dog that is small in number and is used to refer to both old established breeds such as the Stabyhoun and Glen of Imaal Terrier or newer creations.  Since dogs have greater genetic variability than other domesticated animals the number of possible breeds is vast with new crosses constantly occurring, from these both selected and random crosses may come new breeds should offspring reliably breed true to type.  New breeds from the wild such as the Carolina Dog are quite rare compared to attempts at breed creation from man as found in the American Hairless Terrier which sought to exploit a mutation.

Teddy Roosevelt Terrier: The Teddy Roosevelt Terrier is a small to medium-sized American hunting terrier.  Lower-set with shorter legs, more muscular, and heavier bone density than its cousin the American Rat Terrier.  There is much diversity in the history of the Teddy Roosevelt Terrier breed and it shares a common early history with the American Rat Terrier, Fox Paulistinha and Tenterfield Terrier.  It is said the Rat Terrier background stems from the terriers or other dogs that were brought over by early English and other working class immigrants.  Since the breed was a farm, hunting and utility dog there was little to no planned breeding other than breeding dogs with agreeable traits to each other in order to produce the desired work ethic in the dog.  It is assumed that the Feist (dog), Bull Terrier, Smooth Fox Terrier, Manchester Terrier, Whippet, Italian Greyhound, the now extinct English White Terrier, Turnspit dog and or Wry Legged Terrier all share in the Teddy Roosevelt Terrier's ancestry.  These early Ratting Terriers were then most likely bred to the Beagle or Beagle cross bred dogs (for increased scenting ability) and other dogs.  Maximizing the influences from these various breeds provides the modern Teddy Roosevelt Terrier with a keen sense of awareness and prey drive, an acute sense of smell and a very high intellect.  Although they tend to be aloof with strangers they are devoted companion dogs with a strong desire to please and be near their owners side at all times.

Chinese Crested Dog: The Chinese crested dog is a hairless breed of dog.  Like most hairless dog breeds, the Chinese crested comes in two varieties, with and without fur, which are born in the same litter: the Powderpuff and the Hairless.

Airedale Terrier: The Airedale Terrier (often shortened to "Airedale"), also called Bingley Terrier and Waterside Terrier, is a dog breed of the terrier type that originated in the valley ("dale") of the River Aire, in the West Riding of Yorkshire, England.  It is traditionally called the "King of Terriers" because it is the largest of the terrier breeds.  The Airedale was bred from the Old English Black and Tan Terrier (now extinct), the Bull Terrier, the Otterhound and probably some other Terrier breeds, originally to serve as an all around working farm dog.  In Britain this breed has also been used as a war dog, guide dog and police dog.  In the United States, this breed has been used to hunt big game, upland birds, and water fowl, and serve in many other working capacities.

Chow Chow: The Chow Chow (sometimes simply Chow ) is a dog breed originally from northern China, where it is referred to as "Songshi Quan" (Pinyin: sōngshī quǎn 鬆獅犬), which means "puffy-lion dog".  The breed has also been called the "Tang Quan", "Dog of the Tang Empire."  It is believed that the Chow Chow is one of the native dogs used as the model for the Foo dog, the traditional stone guardians found in front of Buddhist temples and palaces.  It is one of the few ancient dog breeds still in existence in the world today.

Terrier Group: Terrier Group is the name of a breed Group of dogs, used by kennel clubs to classify a defined collection of dog breeds.  In general, a "Terrier Group" includes one particular type of dog, the Terrier, although other types may be included in a kennel club's "Terrier Group".  Most major English-language kennel clubs include a "Terrier Group" although different kennel clubs may not include the same breeds in their "Terrier Group".  The international kennel club association, the Fédération Cynologique Internationale, includes Terriers in Group 3 "Terrier", which is then further broken down into four "Sections" based on the type of terrier and breed history.

Mexican Hairless Dog: The Xoloitzcuintli ( ; Nahuatl pronunciation: ), or Xolo for short, is a hairless breed of dog, found in toy, miniature, and standard sizes.  The Xolo also comes in a coated variety and coated and hairless can be born in the same litter.  It is also known as Mexican hairless dog in English-speaking countries, and is one of several breeds of hairless dog.

Fox Terrier: Fox Terriers are two different breeds of the terrier dog type: the Smooth Fox Terrier and the Wire Fox Terrier.  Both of these breeds originated in the 19th century from a handful of dogs who are descended from earlier varieties of British terriers, and are related to other modern white terrier breeds.  In addition, a number of breeds have diverged from these two main types of fox terrier and have been recognised separately, including the Jack Russell Terrier, Miniature Fox Terrier and Rat Terrier.  The Wire and Smooth Fox Terriers share similar characteristics, the main differences being in the coat and markings.  They have been successful in conformation shows, more prominently in America than their homeland.

Vulnerable Native Breeds: Vulnerable Native Breeds are a group of dog breeds originating in the United Kingdom and Ireland, and identified by The Kennel Club (KC) as having annual registration numbers of 300 puppies or fewer.  The need for such a list was first identified in June 2003, with research conducted by the KC to identify the extent of the vulnerability and viability of each breed.  It was a joint project, with the KC working with the British and Irish Native Breeds Trust, later to be known simply as the Native Dog Breeds Trust.  The breeds on the list have been promoted at events such as Discover Dogs and Crufts, and by asking that owners of these breeds mate their dogs rather than having them spayed.

American Hairless Terrier: The American Hairless Terrier is a rare breed of dog that was derived as a variant of Rat Terrier.  As of January 1, 2004, the United Kennel Club deemed the AHT a separate terrier breed, granting it full UKC recognition.  An intelligent, social and energetic working breed, the American Hairless Terrier is often listed as a potential good breed choice for allergy sufferers.

Supporting paragraphs: Chow Chow American Hairless Terrier
Answer: yes