Angles of a triangle

In triangle ABC, the angle beta is 15° greater than the angle alpha. The remaining angle is 30° greater than the sum of the angles alpha and beta. Calculate the angles of a triangle.

Final Answer:

α =  30 °
β =  45 °
γ =  105 °

Step-by-step explanation:

$$\begin{gathered} \beta =15+\alpha \\ \gamma =30+\alpha +\beta \\ \alpha +\beta +\gamma =180 \\ \alpha -\beta =-15 \\ \alpha +\beta -\gamma =-30 \\ \alpha +\beta +\gamma =180 \\ Row2-Row1\to Row2 \\ \alpha -\beta =-15 \\ 2\beta -\gamma =-15 \\ \alpha +\beta +\gamma =180 \\ Row3-Row1\to Row3 \\ \alpha -\beta =-15 \\ 2\beta -\gamma =-15 \\ 2\beta +\gamma =195 \\ Row3-Row2\to Row3 \\ \alpha -\beta =-15 \\ 2\beta -\gamma =-15 \\ 2\gamma =210 \\ \gamma =\frac{210}{2}=105 \\ \beta =\frac{-15+\gamma }{2}=\frac{-15+105}{2}=45 \\ \alpha =-15+\beta =-15+45=30=30^{\circ } \\ \alpha =30 \\ \beta =45 \\ \gamma =105 \end{gathered}$$

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