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Let us consider problem number 21 in the following link

http://www.naec.ge/images/doc/EXAMS/math_2013_ver_1_web.pdf

It is from georgian national exam, it is written (ამოცანა 21), where word "ამოცანა" means amocana or problem. We should find angle $\angle ADE$. I have calculated angle $B$, which is equal to $87^\circ$, but is there any sign of similarity between these two triangle or how can I find it? I think I could calculate angle using arc formula, but I don't remember exactly how it is, even how can I connect arc's angle and $\angle ADE$ angle together? Please help me.

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I think that should be $87^\circ$ for angle $\angle ABC$. –  dtldarek Jul 15 '13 at 9:44
    
aa yes yes right 87 –  dato datuashvili Jul 15 '13 at 9:45

2 Answers 2

up vote 2 down vote accepted

Hint:

  • A convex quadrilateral BCDE is cyclic if and only if its opposite angles sum up to $180^\circ$.

See also the Wikipedia.

I hope this helps ;-)

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so it means that angle EDC is equal to $180-87=93$ and angle ADE is equal to $87$? –  dato datuashvili Jul 15 '13 at 9:48
    
@dato So it seems ;-) –  dtldarek Jul 15 '13 at 9:49
    
thanks a lot of –  dato datuashvili Jul 15 '13 at 9:53

Lets say $\widehat{ED}=\alpha$ and $\widehat{BC}=\beta$. We know that $36=\frac{\beta-\alpha}{2}$. And also $\widehat{EDC}=174^\circ$. So we have $\widehat{EB}=114-\alpha$. Hence $\widehat{EBC}=114+\beta-\alpha$, i.e.$\widehat{EBC}=114+72=186^\circ$. Thus, $\angle EDC=93^\circ$ and $\angle EDC=87^\circ$.

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