Step 1:
From the figure
[tex]\begin{gathered} x\text{ + 129 = 180 \lparen sum of angles on a straight line\rparen} \\ \\ x\text{ = 180 - 129} \\ \\ x\text{ = 51} \end{gathered}[/tex]
Step 2
[tex]\begin{gathered} sin\text{ x = }\frac{Opposite}{Hypotenuse} \\ sin51\text{ = }\frac{h}{8} \\ \\ 0.777\text{ = }\frac{h}{8} \\ h\text{ = 8 }\times\text{ 0.777} \\ \\ h\text{ = 6.22cm} \end{gathered}[/tex]
Step 3
[tex]\begin{gathered} Area\text{ of a triangle = }\frac{1}{2}\text{ }\times\text{ base }\times\text{ h} \\ \\ =\text{ }\frac{1}{2}\text{ }\times\text{ 6.22 }\times8 \\ \\ Area\text{ of a triangle = 24.9 cm}^2 \end{gathered}[/tex]
Step 4
[tex]\begin{gathered} Area\text{ of a sector = }\frac{\theta}{360}\text{ }\times\pi r^2 \\ \\ =\text{ }\frac{129}{360}\text{ }\times\text{ }\frac{22}{7}\text{ }\times\text{ 8}^2 \\ \\ =\text{ 72.08 cm}^2 \end{gathered}[/tex]
Area of the shaded part = 72.08 - 24.9 = 47.18
Final answer
[tex]\begin{gathered} Angle\text{ x = \lbrack51\rbrack} \\ \\ Area\text{ of the shaded part = 47.2 cm}^2 \end{gathered}[/tex]
