Question
Which angle is greater: $\angle XOY$ or $\angle A O B$ ? Give reasons.
$\begin{array}{l}\text {[HINT: } \angle XOY =\angle XOA +\angle A O Y \\ \angle A O B=\angle B O Y+\angle A O Y \\ \therefore \angle XOY -\angle A O B=[\angle X O A+\angle A O Y]-[\angle B O Y+\angle A O Y]\\ =\angle X O A-\angle B O Y\end{array}$
We observe that $\angle X O A>\angle B O Y$
$
\therefore \angle X O A-\angle B O Y>0 \Rightarrow \angle X O Y-\angle A O B>0 \Rightarrow \angle X O Y>\angle A O B]
$
$\begin{array}{l}\text {[HINT: } \angle XOY =\angle XOA +\angle A O Y \\ \angle A O B=\angle B O Y+\angle A O Y \\ \therefore \angle XOY -\angle A O B=[\angle X O A+\angle A O Y]-[\angle B O Y+\angle A O Y]\\ =\angle X O A-\angle B O Y\end{array}$
We observe that $\angle X O A>\angle B O Y$
$
\therefore \angle X O A-\angle B O Y>0 \Rightarrow \angle X O Y-\angle A O B>0 \Rightarrow \angle X O Y>\angle A O B]
$