Oblique Collisions and Momentum Conservation in Identical Mass Objects

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In an oblique collision (when one object hits another object of IDENTICAL mass at a slight angle the two objects move at right angles to each other.

my book says

p2 = p12 + p22

For momenum conservtation:

p= p1+ p2

apparently these equations can only be consistent of the collision occurs at right angles. why?
 
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These are vectors, right?
p.p = (p1+p2).(p1+p2) [2nd equn]
= p1.p1 + 2.p1.p2 + p2.p2
= p.p + 2.p1.p2 [1st equn]
Hence p1.p2 = 0
 
haruspex said:
These are vectors, right?
p.p = (p1+p2).(p1+p2) [2nd equn]
= p1.p1 + 2.p1.p2 + p2.p2
= p.p + 2.p1.p2 [1st equn]
Hence p1.p2 = 0

where has p2.p2 gone in the final line
surely the final line should be
= p1.p2 + p2.p2, no?
 
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1st equn says p.p = p1.p1 + p2.p2
To get from my second line to my third line, use this to substitute p.p for p1.p1 + p2.p2.