1. **Problem statement:** We are given triangle $\triangle EFG$ with incenter $S$, where angle bisectors $ES$, $FS$, and $GS$ meet. Points $P$, $Q$, and $R$ are the feet of perpendiculars from $S$ to sides $EF$, $FG$, and $EG$ respectively. Given: $PS=16$, $ES=21$, $m\angle PER=92^\circ$, and $m\angle RGS=18^\circ$. We need to find $m\angle RGQ$, $m\angle QFS$, and length $QS$. 2. **Key facts and formulas:** - The incenter $S$ is equidistant from all sides, so $SP=SQ=SR$ (all perpendicular distances from $S$ to sides). - Since $P$, $Q$, and $R$ are feet of perpendiculars, $SP=16$ implies $SQ=16$. - Angles involving points on the triangle and incenter relate to angle bisector properties. 3. **Find $m\angle RGQ$:** - $R$ lies on $EG$, $Q$ lies on $FG$, and $G$ is vertex. - Given $m\angle RGS=18^\circ$, and $S$ lies on angle bisector $GS$, so $\angle RGS$ is part of the angle at $G$. - Since $Q$ lies on $FG$, and $R$ on $EG$, $\angle RGQ$ is the angle at $G$ between points $R$ and $Q$. - Because $S$ is on bisector $GS$, $\angle RGQ = 2 \times m\angle RGS = 2 \times 18^\circ = 36^\circ$. 4. **Find $m\angle QFS$:** - $F$ is vertex, $Q$ lies on $FG$, and $S$ is incenter. - Given $m\angle PER=92^\circ$, and $P$ lies on $EF$, $E$ vertex. - Since $ES$ is angle bisector at $E$, and $FS$ is angle bisector at $F$, angles at $F$ relate to $QFS$. - By angle bisector properties and triangle angle sum, $m\angle QFS = 90^\circ - \frac{m\angle PER}{2} = 90^\circ - 46^\circ = 44^\circ$. 5. **Find length $QS$:** - Since $S$ is incenter, $SQ$ is perpendicular distance from $S$ to side $FG$. - Given $SP=16$, and $SP=SQ=SR$, so $QS=16$. **Final answers:** $$ m\angle RGQ = 36^\circ $$ $$ m\angle QFS = 44^\circ $$ $$ QS = 16 $$