Given: δabc is a right triangle.
prove: a2 + b2 = c2

right triangle bca with sides of length a, b, and c. perpendicular cd forms right triangles bdc and cda. cd measures h units, bd measures y units, da measures x units.

the following two-column proof with missing justifications proves the pythagorean theorem using similar triangles:

statement justification
draw an altitude from point c to line segment ab
let segment bc = a
segment ca = b
segment ab = c
segment cd = h
segment db = y
segment ad = x
y + x = c
c over a equals a over y and c over b equals b over x
a2 = cy; b2 = cx
a2 + b2 = cy + b2
a2 + b2 = cy + cx
a2 + b2 = c(y + x)
a2 + b2 = c(c)
a2 + b2 = c2

which is not a justification for the proof?
pieces of right triangles similarity theorem
side-side-side similarity theorem
substitution
addition property of equality