First of all, let us find the value of x
Notice that the sum of central angles 4x and (2x+24) must be equal to 180° (half of the entire circle)
[tex]\begin{gathered} 4x+(2x+24)=180 \\ 6x+24=180 \\ 6x=180-24 \\ 6x=156 \\ x=\frac{156}{6} \\ x=26 \end{gathered}[/tex]
So, the value of x is 26
The arc UY is given by
[tex]mUY=2x+24=2(26)+24\degree=52+24\degree=76\degree[/tex]
So, the arc UY is 76°
The arc VW must be equal to the arc UY since their central angles are vertically opposite angles.
[tex]mVW=76\degree[/tex]
So, the arc VW is 76°
The arc WX must be half of the arc UV
[tex]mWX=\frac{mUV}{2}=\frac{4x}{2}=\frac{4(26)}{2}=\frac{104}{2}=52\degree[/tex]
So, the arc WX is 52°
Finally, the arc WUY is given by
[tex]\begin{gathered} mWUY=360\degree-4x \\ mWUY=360\degree-4(26)_{} \\ mWUY=360\degree-104\degree \\ mWUY=256\degree \end{gathered}[/tex]
So, the arc WUY is 256°
Therefore, the arcs are
[tex]\begin{gathered} x=26 \\ arc\; UY=76\degree \\ arc\; VW=76\degree \\ arc\; WX=52\degree \\ arc\; WUY=256\degree \end{gathered}[/tex]
