Solution Exercise 2.38(b).  
Proof is an idol before whom the pure mathematician tortures himself.
—Arthur Stanley Eddington (1882–1944)

Exercise 2.38. (b) Prove the existence of two lines perpendicular to each other.
 
Proof. By Postulate 5(a), there exist three noncollinear points A, B, and C. By Postulate 1, there is exactly one line AB that contains points A and B. By the Plane Separation Postulate, line AB determines a half-plane containing point C. Thus by the Angle Construction Postulate, there is a unique ray AP with P and C on the same side of line AB such that the measure of angle BAP is 90. By the definition of a right angle, angle BAP is a right angle. Thus, line AB is perpendicular to line AP. Therefore, there exist two lines perpendicular to each other.//