Consider the following definition Definition 358 Two lines l

Consider the following definition. Definition 3.5.8: Two lines l and m are perpendicular if there exists a point A that lies on both l and m and there exists points B elementof l and C elementof m such that BAC is a right angle (ie., mu (BAC) = 90 degree).Notation: l m. Fill in the blanks of the given theorem\'s incomplete proof. For each blank, choose the postulate that best justifies the corresponding sentence. Theorem 3.5.9: If l is a line and P elementof l, then there exists exactly one line m such that p elementof m and m l. Proof: Let l be a line, and let P elementof (hypothesis). Infinitely many points lie on l (a). So there exists a point Q elementof l such that Q notequalto P. Let H be one of the two half-planes bounded by l (b). Then there exists a unique ray PE such that E elementof H and u(QPE) = 90 degree (c). So there exists exactly one line m such that P elementof m and E element m and E elementof m (d). Therefore l m....

Solution

(a) Let l and m be two distinct, nonparallel lines. By definition of nonparallel there exists at least one point, P, that lies on both l and m. Suppose there exists a different point, Q, be on both l and m. By Axiom, there exists exactly one line on which the two distinct points lie. Since P and Q are on both l and m, then l and m are the same line. But l and m are defined as being distinct lines which is a contradiction. Thus, there exists exactly one point P such that P lies on both l and m.

(b) Let l be a line and H be a half-plane bounded by l. We need to show that that H l. Let points A, B exist such that we construct AB. We consider 3 cases, when AB H, when AB l and when A l and B H. When AB H by the Plane Separation Postulate AB is in H l. When AB l by the Incidence Postulate they are in H l. Let A l and B H so AB H by the Ray Theorem, therefore H l. Thus H l is convex.

(d)  Let l be a line where P and Q are two distinct points on l. By the Angle Construction Postulate, there exists a unique ray PA such that A is in one half-plane bounded by l and µ(AP Q) = 90 . Then, we can extend ray PA to a line by the Incidence Postulate. So, there exists exactly one line PA such that P lies on P A and P A l.