1. D

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2. To prove an "if-then" we start by assuming that everything in the "if" part is true.  We then try to make legal changes to things or get things by combining things, and we do this until we get to the "then" part.

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We have:

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a=b

c=b

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Using "Symmetry of Equality, which states that if a first thing equals a second thing, then the second thing equals the first thing, we can take 

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c=b and make b=c.

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We now have a=b and b=c and with Transitivity of Equality, which states that if a first thing is equal to a second thing, and a second thing is equal to a third thing, then the first thing equals the third thing, we have

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a=c.

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We have shown that we can work from a=b and c=b to a=c, thus proving Euclid's Common Notion One using Symmetry of Equality and Transitivity of Equality.

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3.

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4.  We start with the equation given and then algebraic manipulations to get to something that we declare to be obviously false, such as a=b.

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a+(b*c) = (a+b)*(a+c)

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a+b*c = a^2 + ab + ac + bc

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a = a^2 + ab + ac

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1 = a + b + c 

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An axiom requires that the statement is true for any set of values a,b,c.   Here we see that once two of the variables are declared, the other variable is constrained.

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5.

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