Question Number 79368 by TawaTawa last updated on 24/Jan/20
Commented by TawaTawa last updated on 24/Jan/20
$$\mathrm{31}\:+\:\mathrm{105}\:+\:\angle\mathrm{Q}\:\:\:=\:\:\mathrm{180}, \\ $$$$\Rightarrow\:\:\:\angle\:\mathrm{Q}\:\:\:=\:\:\mathrm{44} \\ $$$$\therefore\:\:\:\:\:\:\angle\:\mathrm{S}\:\:+\:\:\angle\mathrm{Q}\:\:\:\:=\:\:\mathrm{180},\:\:\:\:\:\:\:\:\:\angle\mathrm{S}\:\:=\:\:\mathrm{136} \\ $$$$ \\ $$$$\angle\mathrm{P}\:+\:\angle\mathrm{R}\:\:=\:\:\mathrm{180},\:\:\:\:\:\:\:\angle\mathrm{R}\:\:=\:\:\mathrm{180}\:−\:\mathrm{150}\:\:=\:\:\mathrm{30}\:\:\:\left(\mathrm{confuse}\:\mathrm{here}\right). \\ $$
Commented by mr W last updated on 24/Jan/20
$${wrong}! \\ $$$$\angle{PQR}\neq\mathrm{105}° \\ $$$$\angle{Q}\:{is}\:{much}\:{bigger}\:{than}\:\mathrm{44}°\:!\:{you}\:{can} \\ $$$${see}\:{this}\:{in}\:{the}\:{figure}. \\ $$
Commented by TawaTawa last updated on 25/Jan/20
$$\mathrm{Sir},\:\mathrm{in}\:\mathrm{question}.\:\:\angle\mathrm{QPS}\:\:=\:\:\mathrm{150} \\ $$
Commented by mr W last updated on 25/Jan/20
$${but}\:{in}\:{your}\:{equation} \\ $$$$\mathrm{31}\:+\:\mathrm{105}\:+\:\angle\mathrm{Q}\:\:\:=\:\:\mathrm{180}, \\ $$$${you}\:{mean}\:{with}\:\mathrm{105}°\:{the}\:{angle}\:\angle{PQR}! \\ $$$${which}\:{angle}\:{together}\:{with}\:\mathrm{31}°\:{and}\:\angle{Q} \\ $$$${makes}\:\mathrm{180}°\:?! \\ $$
Commented by mr W last updated on 25/Jan/20
$$\angle{R}=\mathrm{31}+{y} \\ $$$${how}\:{can}\:\angle{R}=\mathrm{30}°\:?! \\ $$
Commented by TawaTawa last updated on 25/Jan/20
$$\mathrm{That}\:\mathrm{was}\:\mathrm{why}\:\mathrm{am}\:\mathrm{comfused}. \\ $$$$\mathrm{But}\:\mathrm{i}\:\mathrm{understand}\:\mathrm{now}\:\mathrm{it}\:\mathrm{is}\:\mathrm{typo}\:\mathrm{error}\:\mathrm{in}\:\mathrm{question}. \\ $$
Answered by key of knowledge last updated on 24/Jan/20
$$\angle\mathrm{P}+\angle\mathrm{R}=\frac{\mathrm{P}\overset{\frown} {\mathrm{R}Q}+\mathrm{Q}\overset{\frown} {\mathrm{P}S}}{\mathrm{2}}=\frac{\mathrm{360}}{\mathrm{2}}=\mathrm{180} \\ $$$$\Rightarrow\mathrm{105}+\left(\mathrm{31}+\mathrm{y}\right)=\mathrm{180}\Rightarrow\mathrm{y}=\mathrm{44} \\ $$$$\mathrm{x}:\mathrm{not}\:\mathrm{enouth}\:\mathrm{informion} \\ $$
Commented by mr W last updated on 24/Jan/20
$${check}\:{the}\:{figure}\:{sir}.\:{i}\:{think}\:{x}=\mathrm{31}°. \\ $$
Answered by mr W last updated on 25/Jan/20
$$\angle{PSQ}=\angle{PRQ} \\ $$$$\Rightarrow{x}=\mathrm{31}° \\ $$$$ \\ $$$$\angle{QPS}+\angle{QRS}=\mathrm{180}° \\ $$$$\mathrm{105}+\mathrm{31}+{y}=\mathrm{180}° \\ $$$$\Rightarrow{y}=\mathrm{180}−\mathrm{105}−\mathrm{31}=\mathrm{44}° \\ $$
Commented by TawaTawa last updated on 25/Jan/20
$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir}.\:\mathrm{I}\:\mathrm{understand}\:\mathrm{now}.\:\mathrm{I}\:\mathrm{appreciate}\:\mathrm{your}\:\mathrm{work}\:\mathrm{sir}. \\ $$