The inscribed angle SKQ intercepts the arc SQ. The next equation relates their measures:
[tex]\begin{gathered} \angle SKQ=\frac{1}{2}\hat{SQ} \\ 75=\frac{1}{2}\hat{SQ} \\ 75\cdot2=\hat{SQ} \\ 150\text{ \degree}=\hat{SQ} \end{gathered}[/tex]
Arc SQ can be expressed as the addition of arcs SR and RQ.
[tex]\begin{gathered} \hat{SQ}=\hat{SR}+\hat{RQ} \\ 150=\hat{SR}+60 \\ 150-60=\hat{SR} \\ 90\text{ \degree}=\hat{SR} \end{gathered}[/tex]
The inscribed angle KQR intercepts the arc KR. The next equation relates their measures:
[tex]\begin{gathered} \angle KQR=\frac{1}{2}\hat{KR} \\ 78=\frac{1}{2}\hat{KR} \\ 78\cdot2=\hat{KR} \\ 156\text{ \degree}=\hat{KR} \end{gathered}[/tex]
Arc KR can be expressed as the addition of arcs KS and SR.
[tex]\begin{gathered} \hat{KR}=\hat{KS}+\hat{SR} \\ 156=\hat{KS}+90 \\ 156-90=\hat{KS} \\ 66\text{ \degree}=\hat{KS} \end{gathered}[/tex]
Finally,
[tex]\begin{gathered} \hat{KS}=11x \\ 66=11x \\ \frac{66}{11}=x \\ 6=x \end{gathered}[/tex]
