why is it that since triangle PAB scales by 9/15 to triangle PCD that its sides also scale by 9/15?

How do we know that angles PAB and PCD are congruent here? 
Also, later at around timestamp 8:55 of the video, why is it that since triangle PAB scales by 9/15 to triangle PCD that its sides also scale by 9/15? Why not sqrt 9/15 since triangles have area while lines don't?

Hello Xiao Ma,
We know that DC and AB are parallel, and thus angle PAB and angle PCD are alternating interior angles. So they are equivalent!
Your second question is an especially good one! We see this a lot in geometry: scaling lengths by some factor "r" makes the area scale by "r squared", or the volume by "r cubed."
But in this problem, the 9/15 is referring to how the lengths are scaled, not the areas. So we don't need to worry about squaring or square rooting. Hopefully, this clears up some confusion.Happy Learning,
The Daily Challenge Team

But what about the slant in the shape of the parallelogram? Does that make a difference?

Hi Xiao Ma,
Given what we know about trapezoids (namely that one pair of sides is perpendicular), it doesn't matter if the left and right sides aren't parallel. We just need to know that the top and bottom are parallel.
Happy Learning,
The Daily Challenge Team