What is the explanation for the strange angle result in this trapezoid problem?

[Femme_physics]

Homework Statement

http://img69.imageshack.us/img69/281/trap01.jpg

AC and AC both equal 13. The given angles are in the pic.

The Attempt at a Solution

Attached. Why do I get angle ADC 70? It doesn't make visual sense, and I was told by a very smart person to always trust my visual cues! So I know for sure I've made a mistake somewhere, I'm just not sure where.

 

[Slats18]

Hi femme_physics =)

Angle ADC = 70 degrees IF the shape is a parallelogram, which would mean DAC is a right angle. As this is obviously a trapezoid, I suggest you don't follow your visual cue here =)

I don't remember much of the method of finding angles of non-right angled triangles, but I vaguely remember needing the length of 2 sides and an angle, or something like that, possibly the law of cosines?? Sorry I can't be of more help (if I helped at all :P)

 

[Femme_physics]

Are you telling me that angle BAC does not equal angle ACD?

 

[Slats18]

Not at all, I'm stating that the angle of DAC would be a right angle if it was a parellelogram, aka your answer of 70 degrees. I think you're right about those two angles being the same, again, don't know much about similar triangles as I haven't done them for years. Assuming you are right, you now have an angle (ACD) in the scalene triangle, the top half of the shape. You're already given one side (AD), and you can find the length of AC from the right angled triangle, so now I think you can use the law of cosines to find the length of the CD and then re-input these to find your desired angle =)

 

[Femme_physics]

Not at all, I'm stating that the angle of DAC would be a right angle if it was a parellelogram, aka your answer of 70 degrees

Yes, that's what confuses me! The result shows 70 degrees, the visual does not.

Assuming you are right

Wait, if this is right then why isn't the rest of my calculation right? Law of Sines seems perfectly legitimate way to do it.

 

[I like Serena]

Hi FP,

Are you telling me that angle BAC does not equal angle ACD?

These angles are the same.


Your problem is that sin is a funny function.
Did you know that sin(70) = sin(110)?

 

[Femme_physics]

Did you know that sin(70) = sin(110)?

!

AHA!

ILS to the rescue! :DSo friggin' tricky! I knew that each angle over 90 has its 0-90 correspondent angle, it just didn't occur to me! Would've never come up with it on my own. Thank you. I will keep trusting my visual cues :)

 

[Slats18]

Whoops, completely forgot that fact, should've been the first thing haha, thanks I like Serena =)

 

[I like Serena]

Thank you!
(Although I feel slightly undeserving for such gratitude with only a small remark.)

Very visually acute!