Quadrilateral 81137

In a quadrilateral, angle α is twice as large as angle β. Angle γ is 80% of angle α, and angle δ is 30° greater than angle α. Determine the angles of this quadrilateral.

Correct answer:

α =  68.75 °
β =  137.5 °
γ =  55 °
δ =  98.75 °

Step-by-step explanation:

$$\begin{gathered} \alpha +\beta +\gamma +(30+\alpha )=360 \\ \beta =2\cdot \alpha \\ \gamma =0.80\cdot \alpha \\ 2\alpha +\beta +\gamma =330 \\ 2\alpha -\beta =0 \\ 0.8\alpha -\gamma =0 \\ Row2-Row1\to Row2 \\ 2\alpha +\beta +\gamma =330 \\ -2\beta -\gamma =-330 \\ 0.8\alpha -\gamma =0 \\ Row3-\frac{0.8}{2}\cdot Row1\to Row3 \\ 2\alpha +\beta +\gamma =330 \\ -2\beta -\gamma =-330 \\ -0.4\beta -1.4\gamma =-132 \\ Row3-\frac{-0.4}{-2}\cdot Row2\to Row3 \\ 2\alpha +\beta +\gamma =330 \\ -2\beta -\gamma =-330 \\ -1.2\gamma =-66 \\ \gamma =\frac{-66}{-1.2}=55 \\ \beta =\frac{-330+\gamma }{-2}=\frac{-330+55}{-2}=137.5 \\ \alpha =\frac{330-\beta -\gamma }{2}=\frac{330-137.5-55}{2}=68.75=68.75^{\circ }=68°45^{\prime } \\ \alpha =\frac{275}{4}=68.75 \\ \beta =\frac{275}{2}=137.5 \\ \gamma =55 \end{gathered}$$

$\delta =30+\alpha =30+68.75=98.75^{\circ }=98°45^{\prime }$

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