document.write( "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.\r
\n" ); document.write( "\n" ); document.write( "Theorem 1.1.4. If two lines intersect, then their union lies in exactly one plane.\r
\n" ); document.write( "\n" ); document.write( "Proof: Let and be two intersecting lines.
\n" ); document.write( " a. A ∩ B is a point p.
\n" ); document.write( " b. B contains a point q ≠ p.
\n" ); document.write( "c. There is a plane E, containing A and q.
\n" ); document.write( "d. E contains A ∪ B.
\n" ); document.write( "e. No other plane contains A ∪ B.''. \n" ); document.write( "

Algebra.Com's Answer #827154 by shlomitg(8) $\"\"$ $\"About$

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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.\r
\n" ); document.write( "\n" ); document.write( "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.\r
\n" ); document.write( "\n" ); document.write( "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.\r
\n" ); document.write( "\n" ); document.write( "d. E contains A ∪ B. Plane E contains the two lines, therefore containing A ∪ B.\r
\n" ); document.write( "\n" ); document.write( "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. \n" ); document.write( "

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