Given:
The measure of arc XY is (31x)°
The measure of arc YZ is (35x - 16)°
We need to determine the value of x, measure of arc XYZ and arc XZ.
Value of x:
From the figure, it is obvious that the arcs XY and YZ are congruent.
Thus, we have;
[tex]31x=35x-16[/tex]
Subtracting both sides by 35x, we have;
[tex]-4x=-16[/tex]
[tex]x=4[/tex]
Thus, the value of x is 4.
Measure of arc XYZ:
The measure of arc XYZ is given by
[tex]m \widehat{XYZ}=m \widehat{XY}+m \widehat{YZ}[/tex]
The measure of arc XY is given by
[tex]m \widehat{X Y}=31(4)=124^{\circ}[/tex]
The measure of arc YZ is given by
[tex]m \widehat{Y Z}=35(4)-16=124^{\circ}[/tex]
Hence, the measure of arc XYZ is given by
[tex]m \widehat{XYZ}=124^{\circ}+124^{\circ}=248^{\circ}[/tex]
Therefore, the measure of arc XYZ is 248°
Measure of arc XZ:
The measure of arc XZ is given by
[tex]m \widehat{XZ}=360^{\circ}-m \widehat{XYZ}[/tex]
Substituting the values, we have;
[tex]m \widehat{XZ}=360^{\circ}-248^{\circ}[/tex]
[tex]m \widehat{XZ}=112^{\circ}[/tex]
Thus, the measure of arc XZ is 112°
