Math Problem (Not Kite Related)
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Found a SAT problem that I couldn't figure out.

Without using references I am stymied.

Any of you want to help?

I know that (a+c) = (d+e), but can't get beyond that.

I'm guessing that B is the answer, but don't know why.

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It's been over 25 years, but I'm game.

What do we know:
a+b+c = b+d+e=180 degrees.
Therefore a+c = d+e

Forget it. I don't remember.
Ca Ike:
This one is a bit of a stumper.  Since there are no measurements whatsoever you really have nothing but visual cues to go on and maybe a few simple rules. THis is not a math problem but a logic problem.

THe only things we can know for sure is that a+b+c=180 degrees  as should b+d+e. We can assume that a, b and c are not equal due to the lean of the triangle and that A and C are not equal.  By the wording of the question you can eliminate A,C and d as possible answers so it has to be B or D since they are the only choices with "pairs" of angles. Now since B is the same for both we can surmise that d+e=a+c.

Now IF c=d and d=e then c=e and since we know that c+b+a=180 and d+e+b=180 then we can also surmise that a is equal to c,d and e.  Therefore since answer E does not allow for a to be equal to any of them the answer is "B".  If a and e are equal then a+b=e+b therefore d+b=c+b.  B is the only answer that logically allows for all angles to be considered in the equation.
Will Sturdy:
I asked my brother, who's rather good with this sort of thing. Here's what he said:

There are theorems that the angle between a tangent and a cord is half the subtended arc, as is the inscribed angle. So a=e, c=d.
Proof: Draw a triangle with base db and opposite side the center of the circle, noting that it is an isosceles triangle with central angle the angle of the arc bd and the other angles equal to the complement of a (since a radius and a tangent meet in a right angle). The central angle is thus 180-2comp(a)=180-2(90-a)=2a.
The other (that e is half the angle of the inscribed arc) is a standard theorem, and looks a bit harder to prove.
Quote from: chilese on January 19, 2013, 01:37 PM

I'm guessing that B is the answer, but don't know why.

B is the correct answer.  But I can't tell you why.
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