Kąt β jest kątem wpisanym opartym na tym samym łuku co kąt środkowy α, zatem:

$\alpha =2\beta$

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$\text{a)}$  

$\alpha +\beta ={111}^o$

$2\beta +\beta ={111}^o$

$3\beta ={111}^o$    

$\beta =\frac{{111}^o}{3}$

$\beta ={37}^o$

Wobec tego:

$\alpha =2\beta =2\cdot {37}^o={74}^o$

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$\text{b)}$  

$\alpha =\beta +{56}^{o}30{}^{\prime }$

$2\beta =\beta +{56}^{o}30{}^{\prime }$

$2\beta -\beta ={56}^{o}30{}^{\prime }$

$\beta ={56}^{o}30{}^{\prime }$

Wobec tego:

$\alpha =2\cdot \beta =2\cdot {56}^{o}30{}^{\prime }={113}^o$

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$\text{c)}$  

$\frac{1}{2}\alpha +\frac{1}{3}\beta ={48}^o$

$\frac{1}{2}\cdot 2\beta +\frac{1}{3}\beta ={48}^o$

$\beta +\frac{1}{3}\beta ={48}^o$

$1\frac{1}{3}\beta ={48}^o$ $/:1\frac{1}{3}$

$\beta ={36}^o$

Wobec tego:

$\alpha =2\cdot {36}^o={72}^o$