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Let us consider problem number 21 in the following link

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
up vote 2 down vote accepted


  • 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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