if a ray OP stands on a line QR such that angle POR : QOP=4:5, prove that angle QOP is equal to one-ninth of angle QOR

Dear student

$$\begin{gathered} Given: \\ \frac{\angle POR}{\angle QOP}=\frac{4}{5} \\ \Rightarrow \frac{\angle QOP}{\angle POR}=\frac{5}{4} \\ Apply componendo and dividendo \\ \Rightarrow \frac{\angle QOP+\angle POR}{\angle QOP-\angle POR}=\frac{5+4}{5-4} \\ \Rightarrow \frac{\angle QOP+\angle POR}{\angle QOP-\angle POR}=\frac{9}{1} \\ \Rightarrow \frac{\angle QOR}{\angle QOP-\angle POR}=\frac{9}{1} \\ \Rightarrow \frac{\angle QOP-\angle POR}{\angle QOR}=\frac{1}{9} \\ Please recheck your question , proceeding like this we cannot \\ get desired result \end{gathered}$$
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