Quandaries
and Queries 

Given:Trapezoid ROSE with diagonals RS and EO intersecting at point M Now, my book gives me all the steps; however, one step, I don't understand. I hope you can help. I'll give you the step: They first proved triangle I congruent to triangle II with SAS using an indirect proof. I understand that. But then they say OR is parallel to SE and the reason is," if 2 lines are cut by a transversal and alternate interior angles are congruent, then the lines are parallel."And then they go on to say that a trapezoid only has one pair of parallel sides and we just saw there are 2 pairs of parallel sides if diagonals RS and EO bisect each other so it can't bisect each other. My question is: how do you know what the alternate interior angles are going on? You could say that OS is parallel to RE from that because if 2 lines, OS and RE, are cut by a transversal,RS, and alternate interior angles are congruent, then the lines, SO and RE are parallel. How do you know its the lines SE and OR being cut with transversal RS; thus, it being parallel and contradicting the given that the figure is a trapezoid? 

Hi Abraham, Below I have reproduced the diagram you sent.
Your book gives a proof by contradiction. The authors first assumes that the diagonals do bisect each other. That is they assume that
They then argue, using SAS that the triangles I and II are congruent, these are the triangles ORM and ESM. At this point you need to be somewhat careful as the diagram may be misleading. The assumption is that OM = ME even though they don't look the same in the diagram. Also RM = MS and hence the congruence of the triangles ORM and ESM implies angles ORM and ESM are congruent. Hence RS is a line that cuts OR and ES and the alternate interior angles (ORM and ESR) are congruent. Hence OR is parallel to ES. I don't like this problem. I prefer the definition that "a trapezoid is a quadrilateral with a pair of parallel sides" rather than "a trapezoid is a quadrilateral with exactly one pair of parallel sides". With my definition a parallelogram is a trapezoid and the problem you were given could then be reworded as Given:Trapezoid ROSE with diagonals RS and EO intersecting at point M Harley 

