For a triangle with all three sides given, we can determine the size/measure of one of the angles by using the law of cosines.
The law of cosines is given as follows;
[tex]a^2=b^2+c^2-2bc\cos A[/tex]
Where the variables are;
[tex]A=\theta,a=160,b=90,c=180[/tex]
We can plug in the values into the cosine formula and we'll have;
[tex]\begin{gathered} 160^2=90^2+180^2-2(90)(180)\cos A \\ 25600=8100+32400-32400\cos A \\ 25600=40500-32400\cos A \\ \text{Subtract 40500 from both sides;} \\ -14900=-32400\cos A \\ \text{Divide both sides by -32400} \\ \frac{-14900}{-32400}=\cos A \\ \cos A=0.4598765 \\ \end{gathered}[/tex]
To determine the angle measure of the result above, we now look up the value of arc-cos 0.4598765 on a calculator, and we'll have;
[tex]\begin{gathered} \cos A=0.4598765 \\ A=\cos ^{-1}0.4598765 \\ A=62.62086 \\ \text{Rounded to one decimal point} \\ A=62.6 \end{gathered}[/tex]
ANSWER:
The indicated angle measures 62.6 degrees
