The Question I have been given:
In the figure below, D is a point on segment CE such that segments AD and BE are parallel, and point A is not on segment BC. Line segments AD and BC intersect at point P.
The solution I have worked on and what I have written as my answer:
One might believe angle CAD can equal angle CBE, with some alterations to the figure. Because segments AD and BE are parallel to each other and both intersect the segment CDE, angles ADC and BEC are equal to each other. Similarly, for angles ACD and BCD to be equal to each other, segments AC and BC would have to be parallel to each other. So, angle CAD and angle CBE can be equal to each other if we can alter where point P is. Doing so would mean "moving" the point P and making segments AC and BC parallel. However, doing so would mean that the segment BC would no longer exist. Instead, we can add another point, S. So, the segments BS and AC would be parallel. But wait, segment BC does not exist anymore, so angle CBE does not exist anymore too! So, angle CAD cannot equal angle CBE.
If the reader is still confused, here is another view on how angle CAD can equal angle CBE:
We can split the figure into two triangles, triangle ACD, and triangle BCE. We already know that segments AD and BE are parallel to each other. Both triangles also have at least some part of the segment CDE as part of their triangle. The two triangles are ALMOST similar, but in mathematical terms, the 2 triangles still aren't similar. For angle CAD to equal angle CBE, triangles ACD and BCE would have to be similar triangles. We would have to "move" point P for segments AC and BC to be parallel. Doing so, however, we would have to add another point, let's say S, to the graph. Instead, after moving point P, segments BS and AC would be parallel. Now, the two triangles are similar, right? But doing so, the segment BC doesn't exist anymore, and angle CBE doesn't exist anymore either. So, angle CAD cannot equal angle CBE.
By moving Point P, the segment BC wouldn't exist anymore, right?? And without BC, angle CBE won't exist either, right? So then if BC doesn't exist, and CBE doesn't exist, then CBE can't equal CAD. Then is my solution correct, or is it flawed?
Thank you in advance, to the person who reads this and helps me out! However, I do need a quick answer, so if it is possible I would really appreciate it if someone could answer this quickly. Thank you again!
Angle CAD can be equal to angle CBE. If angles CAD and CBE are not equal, then we can choose a point A' on AD so that angle CA'D is equal to CBE. So we can move point A to A', and then angles CAD and CBE will be equal.