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Let $x$ and $y$ be parallel lines where $x\neq y$.
How do I prove that $y$ is in one of the $1/2$ planes , let's call it $H$ of $x$ ?
How to prove that one of $1/2$ planes of $y$ is contained in $H$.

Any suggestions or comments are welcome.

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    I understand. I called my teacher(no response) and then went to my school to ask them for tutors early in the semester, but the unfortunate issue is that there is no one available right now that will offer tutoring past Calculus. I wish there was a tutoring company that could offer higher level math tutoring.2012-05-14

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Start off with $P_1,P_2$ as the half-planes of $x$, $H_1,H_2$ the half-planes of $y$. You want to show that $y$ lies in one of $P_1,P_2$, and that one of $H_1,H_2$ is contained in whichever of $P_1,P_2$ that $y$ lies in.

Are you familiar with proofs by contrapositive or contradiction? I'm going to recommend a contradiction approach for the first part, here.

For the first part, let's suppose that $y$ doesn't lie (fully) in either of $P_1,P_2$, so since $y$ and $x$ don't coincide, then a part of $y$ lies in each of $P_1,P_2$. In particular, there is some point $A$ on $y$ in $P_1$, and some point $B$ on $y$ in $P_2$, yes? What does Postulate 9 say about the segment $AB$ and the line $x$, then? What does that mean about the lines $y$ and $x$, since $AB$ is a segment of $y$, and since $y$ and $x$ aren't the same line? We should have the desired contradiction, here.

For the second, go ahead and assume that $y$ lies in $P_1$ (we've shown it lies in one of $P_1,P_2$, and we can always reindex, if need be), and that $H_2$ isn't contained in $P_1$. All you've got to show is that $H_1$ is contained in $P_1$. I'd start by showing that $x$ must lie in $H_2$--by reasoning as in the first part, $x$ must lie in one of $H_1,H_2$, and we can use our assumption that $H_2$ isn't contained in $P_1$ to show that $x$ must lie in $H_2$. After that, recall that all these half-planes are convex, as is any line in the plane, and note that the overlap of two convex sets is again convex. It will be sufficient (since $x$ lies in $H_2$ and $y$ lies in $P_1$) to show that $H_1$ and $P_2$ have no overlap. Hopefully, that's enough to get you going.

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    Thanks for your help, Cameron.2012-05-16
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If the half-planes, H1 and H2, are determined by line y. Let point A be in H1 and point B lies in H2. Select points C and D to lie on line y. Should I prove that line y intersects line segment AB to show that A lies in H1 and B lies in H2? And then how do I show that point B is located in a half plane contained by the one that has point A is on the same half plane. Any suggestions or comments are welcome.

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    Apologies for the delay. I had to go to class.2012-05-15