Question 1193017: 5.Prove theorem 1.1.4. The steps in the proof are already given: you just have to supply the reasons for each step.
Theorem 1.1.4. If two lines intersect, then their union lies in exactly one plane.
Proof: Let and be two intersecting lines.
a. A ∩ B is a point p.
b. B contains a point q ≠ p.
c. There is a plane E, containing A and q.
d. E contains A ∪ B.
e. No other plane contains A ∪ B.''.
Answer by shlomitg(8)   (Show Source):
You can put this solution on YOUR website! a. A ∩ B is a point p. We want to create a plane that contain one of the lines and a point on the other line, different from the intersection since the plane will be defined by a line and point not on the line. Therefore we define p.
b. B contains a point q ≠ p. Here we create a point on the line B which is not on the line A. we prepare the basis for a plane: line A and the point q not on A.
c. There is a plane E, containing A and q. We create the plane which contains lines A and B since it contains line A and the points p and q.
d. E contains A ∪ B. Plane E contains the two lines, therefore containing A ∪ B.
e. No other plane contains A ∪ B. We need to prove that E is the only plane containing lines A and B. All the points q we could have chosen, lie on the line B and they define with line A a one and only plane.
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