Stumped on a geometry question in the Princeton Review

[alsaxon]

I do understand you can assume nothing from the diagrams. My question is on the principles.

Here is the question which I've inserted in three photos
part 1: https://docs.google.com/file/d/0B8G2c7V ... sp=sharing

part 2: https://docs.google.com/file/d/0B8G2c7V ... sp=sharing

part 3: https://docs.google.com/file/d/0B8G2c7V ... sp=sharing

My answer would have easily been C - that m and n are equal (the answer is D). I didn't assume that based on the pic. I assumed that because line AD and DC are equal. I thought that equal-sized lines had equal-sized angles. I'm getting this from all the rules on triangles and angles. For instance here:

https://docs.google.com/file/d/0B8G2c7V ... sp=sharing

Can anyone help me get this straight in my head? I feel like a lot of this stuff is just memorization, but I'd like to make valid connections.

[tommywallach]

Hey Alsaxon,

Indeed, you fell for the classic trap here!

Here are the rules:

Equal sides are opposite equal angles, but only in a single triangle. What you're mistaking is that these are two separate triangles here, and remember: a single angle tells you NOTHING about the length of the line that is opposite it.

Think of this example. Imagine two lines that make a one degree angle. If those lines were very short, the third side of the triangle you might create would be very short. But if those lines were a mile long, the third side of that triangle would actually be pretty long!

In this case, you know that AD and DC are equal. But if you were to grab hold of the top of this triangle (point B) and drag it to the left and the right, or up and down, the angles would keep changing, even though AD and DC would still be as equal as they ever were!

Hope that helps!

-t