I'm struggling to visualize the following GRE problem:
The solution says that for $m$ and $n$ to intersect on the right, it has to be the case that $2x+3x>180$, after that we get $x>36$. But I don't understand why. I can see that if $m$ and $n$ are parallel then $2x+3x = 180$ though. Can someone help me visualize this problem? Thanks.
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This is the classic example of Euclid's fifth postulate, the parallel postulate.
Here, we talk about the angles on the opposite side to which the lines intersect with each other. Hence, their sum must be greater than $180^\circ$.
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The fact that $m $ and $n $ intersect to the right means that $2x+3x=5x>180^0$... For if they were parallel, the sum would equal $180^0$. Intersecting on the right means either $2x $ or $3x $ is larger than in the parallel case...
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The following diagram helps to visualise the solution to your problem.
The lines g and f intersect to the right of the picture. The line h cuts across f and g. The line i is parallel to g, and consequently, $\alpha=\beta$ in the diagram. The situation is similar to the one that you have. By changing the way that h cuts across g and f you can obtain an identical one to your problem.
Regardless, we see that $\gamma+\alpha=\gamma+\beta>180^{\circ}$.
If $m$ and $n$ intersect to the right, then a triangle is formed, two of whose angles are the angles just to the right of those labelled $2x^\circ$ and $3x^\circ$. These angles are $180-2x$ degrees and $180-3x$ degrees respectively, and so $180-2x+180-3x<180$ since the sum of all three angles in the triangle is $180$ degrees. Rearranging this inequality gives $2x+3x>180$.
An intuitive way to think about it is that if $x=36$, then the two lines are parallel. If you make $x$ larger while keeping the diagonal line fixed, that means you are tilting lines $m$ and $n$ away from each other on the left side, and thus towards each other on the right side. If the lines start parallel and you tilt them towards each other on the right just a little bit, then they will meet. Conversely, if you tilted them in the opposite direction (by decreasing $x$), they would meet on the left instead of on the right. So they meet on the right iff $x>36$.