In the Figure in $$\triangle$$ PQR PT $$\perp$$ QR at T and PS is the bisector of $$\angle QPR$$.If $$\angle PQR=78^\circ$$, and $$\angle TPS = 24^\circ$$ then the measure of $$\angle PRQ$$ is

Given figure 

$$\triangle$$ PQR PT $$\perp$$ QR at T and PS is the bisector of $$\angle QPR$$.If $$\angle PQR=78^\circ$$, and $$\angle TPS = 24^\circ$$

In the $$\triangle $$ PQT 

$$ 90^\circ+ 78 ^\circ+ \angle QPT = 180 ^\circ$$

$$\Rightarrow \angle QPT = 180 ^\circ = 180^\circ - 168^\circ $$

                                       = $$12 ^\circ$$ 

then $$\angle QPS = 12^\circ + 24^\circ $$

                            = $$ 36 ^\circ $$

PS is bisect $$\angle QPR $$

then $$\angle SPR = 36^\circ $$

In $$\triangle PTS , 90^\circ +24 ^\circ + \angle PST = 180^\circ $$

$$\angle PST = 180^\circ - 114^\circ $$

                     = $$66\circ $$

the $$\angle PST  = \angle  PRS  + \angle RPS $$

        $$ 66^\circ = 36 ^\circ = \angle PRQ $$

$$\Rightarrow \angle PRQ = 66^\circ - 36^\circ $$

$$\Rightarrow \angle PRQ = 30^\circ $$ Ans