Pythagoras theorem
Statement 1:
ΔABC ≅ ΔCBD ≅ ΔACD
Reason: Given
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Statement 2:
b/c = y/b; a/x = x/a
Reason: corresponding sides of similar triangles are proportional
(we want to have to have in the next statement that b² = cy; a² = cx
and proportionality is usually represented as fractions, if we observe the figure, the fractions of this statement correspond to the division of similar sides of the triangles)
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Statement 3:
b² = cy; a² = cx
Reason: cross product property
(if we multiply both sides of b/c = y/b by b, we obtain b² = cy, and if we do the same for a/x = x/a we obtain a² = cx, since we are multiplying, it is called product, then, the option that best fit this field is cross product property)
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Statement 4:
a² + b² = cx + cy
Reason: addition property of equality
(we want to prove that a² + b² = c², from the previous statement we can add both equalities so we obtain a² + b² , which is nearer to the conclusion we want to prove)
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Statement 5:
a² + b² = c(x + y)
Reason: factor
(we find the common factor of cx and cy, it is c, then cx + cy = c(x + y))
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Statement 6:
c = x + y
Reason: Segment addition postulate
(we almost have the conclusion in the previous statement except for the (x + y) of the right part of the equality, since in the figure we observe that c = x + y, then we can use it to replace (x + y))
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Statement 7:
a² + b² = c²
Reason: substitution
(we substitute c by (x + y) of the statement 5)
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