@Serpentes said:
@adamsna said:
I didn't assume anything that is unreasonable, I used the transversal law [1] to see that many (if not all) of the angles are indeed 90 degree angles.But what you did with this is wrong. The transversal law states that when a transverse line intersects parallel lines, corresponding angles are equal and that opposite interior angles are equal. There are several angles marked on the diagram as being right angles. But we do not have any marked angles that relate b to either a or c, so we cannot assume parallelity there. Nor do we have any data that relates d to e, so we cannot assume parallelity there.
Your only argument is that b appears to have the same slope as a and c, and that d appears to have the same slope as e. But there is nothing that allows you to make those statements with mathematical rigor. What if the slope varies by an extraordinarily small amount? Say, a factor of 10-8? That's well below the resolution of the displayed diagram, but it still means that those lines would eventually intersect -- that they are not parallel. That's why geometry is about formal proofs and not eyeballing some figure on a test.
Using the laws of congruency for angles formed by tranverse lines intersecting parallel lines in a diagram where we cannot prove that lines are parallel is sloppy -- and incorrect -- geometry.
I do see your point. And you are correct, I was assuming those lines are parallel.
If this was a graduate class, I would agree with you and I think the professor should be smacked, however it is not. Let us propose an argument.
The OP said this was a high school class. The OP did not say what year the class was at, so since this is not an argument of mathematics, can we agree that if we assume that this was a freshman level course in math that this kind of content would be reasonable?
In the real world of mathematics, you don't assume anything - no argument there - but in a classroom you do. Teachers make mistakes lecturing, tests, and homework, which includes leaving out important information. I have even seen the wrong answers in the back of the book before. On a test in a HS classroom would you not put down 4 parallel lines? Or could you actually say that you would have the balls to put down "there are no parallel lines"? If you said the second, at that point in your HS education could you actually say you had enough knowledge to actually write out a formal proof showing that there are parallel lines? I did not. And, another argument I would like to propose is that even if you are right, it is up to the teacher to decide to give you the points or not. Even if you had a solid proof, the teacher can still mark it wrong, and I am sure we all had teachers that were like that. I personally would put down 4 parallel lines if I was in that HS class. If I was in a graduate class, I would instruct the professor that they are missing something.
That is my argument for why I "assumed" that d and e were parallel.
I am not a mathematician, but, having a programmable calculator did add to my laziness.