We know all the sides of the triangle. We need to find the missing angles.
To do this, we can use a formula known as the cosine rule:
[tex]c=\sqrt[]{a^2+b^2-2ab\cos\gamma}[/tex]
Each one of the elements involved in this formula is explained in the next diagram:
Therefore, we can solve the equation above for gamma:
[tex]\begin{gathered} c=\sqrt[]{a^2+b^2-2ab\cos\gamma} \\ \Rightarrow c^2-a^2-b^2=-2ab\cos \gamma \\ \Rightarrow a^2+b^2-c^2=2ab\cos \gamma \\ \Rightarrow\cos \gamma=\frac{a^2+b^2-c^2}{2ab} \end{gathered}[/tex]
Let's set a=8,b=11,c=17, then:
[tex]\cos \gamma=-\frac{104}{176}=-\frac{13}{22}[/tex]
We can then use the arccos function to get the value of gamma:
[tex]\begin{gathered} \cos \gamma=-\frac{13}{22} \\ \Rightarrow\gamma=\arccos (-\frac{13}{22})\approx126.2degree \end{gathered}[/tex]
The only option that has such an angle is the fourth one. Therefore, the answer is the fourth option: 126°, 32°, 22°
