Angles of the triangle

ABC is a triangle. The size of the angles alpha and beta are in a ratio of 4:7. The angle gamma is greater than the angle alpha by a quarter of a straight angle. Determine the angles of the triangle ABC.

Final Answer:

α =  36 °
β =  63 °
γ =  81 °

Step-by-step explanation:

$$\begin{gathered} \alpha :\beta =4:7=4/7 \\ \gamma =\alpha +180/4=\alpha +45{}^{\circ } \\ \alpha +\beta +\gamma =180{}^{\circ } \\ \alpha +(\frac{7}{4}\cdot \alpha )+(\alpha +45)=180 \\ \alpha +(7/4\cdot \alpha )+(\alpha +45)=180 \\ 15\alpha =540 \\ \alpha =\frac{540}{15}=36 \\ \alpha =36=36^{\circ } \end{gathered}$$

$\beta =\frac{7}{4}\cdot \alpha =\frac{7}{4}\cdot 36=\frac{7\cdot 36}{4}=\frac{252}{4}=63^{\circ }$

$\gamma =\alpha +45=36+45=81^{\circ }$

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