Proof: Add a Number to Both Sides

Let's prove the following theorem:

if a + 90 = 180, then a = 90

First we use the Additive Property of Equality to claim that:

a + 90 + -90 = 180 + -90

Then we show that:

left side reduces to a.
right side reduces to 90.

Proof:

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Given

1	a + 90 = 180

Proof Table

#	Claim	Reason
1	(a + 90) + (-90) = 180 + (-90)	if a + 90 = 180, then (a + 90) + (-90) = 180 + (-90) (Additive Property of Equality)
2	90 + (-90) = 0	90 + (-90) = 0 (Fixed Value Statement)
3	(a + 90) + (-90) = a + (90 + (-90))	(a + 90) + (-90) = a + (90 + (-90)) (Associative Property of Addition)
4	a + (90 + (-90)) = a + 0	if 90 + (-90) = 0, then a + (90 + (-90)) = a + 0 (Substitution)
5	a + 0 = a	a + 0 = a (Additive Identity)
6	a + (90 + (-90)) = a	if a + (90 + (-90)) = a + 0 and a + 0 = a, then a + (90 + (-90)) = a (Transitive Property of Equality)
7	(a + 90) + (-90) = a	if (a + 90) + (-90) = a + (90 + (-90)) and a + (90 + (-90)) = a, then (a + 90) + (-90) = a (Transitive Property of Equality)
8	180 + (-90) = 90	180 + (-90) = 90 (Fixed Value Statement)
9	a = 90	if (a + 90) + (-90) = 180 + (-90) and (a + 90) + (-90) = a and 180 + (-90) = 90, then a = 90 (Transitive Property Application 2)

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