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Question Number 79368 by TawaTawa last updated on 24/Jan/20

Commented by TawaTawa last updated on 24/Jan/20

31 + 105 + ∠Q = 180, ⇒ ∠ Q = 44 ∴ ∠ S + ∠Q = 180, ∠S = 136 ∠P + ∠R = 180, ∠R = 180 − 150 = 30 (confuse here).

$$\mathrm{31}\:+\:\mathrm{105}\:+\:\angle\mathrm{Q}\:\:\:=\:\:\mathrm{180}, \\ $$$$\Rightarrow\:\:\:\angle\:\mathrm{Q}\:\:\:=\:\:\mathrm{44} \\ $$$$\therefore\:\:\:\:\:\:\angle\:\mathrm{S}\:\:+\:\:\angle\mathrm{Q}\:\:\:\:=\:\:\mathrm{180},\:\:\:\:\:\:\:\:\:\angle\mathrm{S}\:\:=\:\:\mathrm{136} \\ $$$$ \\ $$$$\angle\mathrm{P}\:+\:\angle\mathrm{R}\:\:=\:\:\mathrm{180},\:\:\:\:\:\:\:\angle\mathrm{R}\:\:=\:\:\mathrm{180}\:−\:\mathrm{150}\:\:=\:\:\mathrm{30}\:\:\:\left(\mathrm{confuse}\:\mathrm{here}\right). \\ $$

Commented by mr W last updated on 24/Jan/20

wrong! ∠PQR≠105° ∠Q is much bigger than 44° ! you can see this in the figure.

$${wrong}! \\ $$$$\angle{PQR}\neq\mathrm{105}° \\ $$$$\angle{Q}\:{is}\:{much}\:{bigger}\:{than}\:\mathrm{44}°\:!\:{you}\:{can} \\ $$$${see}\:{this}\:{in}\:{the}\:{figure}. \\ $$

Commented by TawaTawa last updated on 25/Jan/20

Sir, in question. ∠QPS = 150

$$\mathrm{Sir},\:\mathrm{in}\:\mathrm{question}.\:\:\angle\mathrm{QPS}\:\:=\:\:\mathrm{150} \\ $$

Commented by mr W last updated on 25/Jan/20

but in your equation 31 + 105 + ∠Q = 180, you mean with 105° the angle ∠PQR! which angle together with 31° and ∠Q makes 180° ?!

$${but}\:{in}\:{your}\:{equation} \\ $$$$\mathrm{31}\:+\:\mathrm{105}\:+\:\angle\mathrm{Q}\:\:\:=\:\:\mathrm{180}, \\ $$$${you}\:{mean}\:{with}\:\mathrm{105}°\:{the}\:{angle}\:\angle{PQR}! \\ $$$${which}\:{angle}\:{together}\:{with}\:\mathrm{31}°\:{and}\:\angle{Q} \\ $$$${makes}\:\mathrm{180}°\:?! \\ $$

Commented by mr W last updated on 25/Jan/20

∠R=31+y how can ∠R=30° ?!

$$\angle{R}=\mathrm{31}+{y} \\ $$$${how}\:{can}\:\angle{R}=\mathrm{30}°\:?! \\ $$

Commented by TawaTawa last updated on 25/Jan/20

That was why am comfused. But i understand now it is typo error in question.

$$\mathrm{That}\:\mathrm{was}\:\mathrm{why}\:\mathrm{am}\:\mathrm{comfused}. \\ $$$$\mathrm{But}\:\mathrm{i}\:\mathrm{understand}\:\mathrm{now}\:\mathrm{it}\:\mathrm{is}\:\mathrm{typo}\:\mathrm{error}\:\mathrm{in}\:\mathrm{question}. \\ $$

Answered by key of knowledge last updated on 24/Jan/20

∠P+∠R=((PR^⌢ Q+QP^⌢ S)/2)=((360)/2)=180 ⇒105+(31+y)=180⇒y=44 x:not enouth informion

$$\angle\mathrm{P}+\angle\mathrm{R}=\frac{\mathrm{P}\overset{\frown} {\mathrm{R}Q}+\mathrm{Q}\overset{\frown} {\mathrm{P}S}}{\mathrm{2}}=\frac{\mathrm{360}}{\mathrm{2}}=\mathrm{180} \\ $$$$\Rightarrow\mathrm{105}+\left(\mathrm{31}+\mathrm{y}\right)=\mathrm{180}\Rightarrow\mathrm{y}=\mathrm{44} \\ $$$$\mathrm{x}:\mathrm{not}\:\mathrm{enouth}\:\mathrm{informion} \\ $$

Commented by mr W last updated on 24/Jan/20

check the figure sir. i think x=31°.

$${check}\:{the}\:{figure}\:{sir}.\:{i}\:{think}\:{x}=\mathrm{31}°. \\ $$

Answered by mr W last updated on 25/Jan/20

∠PSQ=∠PRQ ⇒x=31° ∠QPS+∠QRS=180° 105+31+y=180° ⇒y=180−105−31=44°

$$\angle{PSQ}=\angle{PRQ} \\ $$$$\Rightarrow{x}=\mathrm{31}° \\ $$$$ \\ $$$$\angle{QPS}+\angle{QRS}=\mathrm{180}° \\ $$$$\mathrm{105}+\mathrm{31}+{y}=\mathrm{180}° \\ $$$$\Rightarrow{y}=\mathrm{180}−\mathrm{105}−\mathrm{31}=\mathrm{44}° \\ $$

Commented by TawaTawa last updated on 25/Jan/20

God bless you sir. I understand now. I appreciate your work sir.

$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir}.\:\mathrm{I}\:\mathrm{understand}\:\mathrm{now}.\:\mathrm{I}\:\mathrm{appreciate}\:\mathrm{your}\:\mathrm{work}\:\mathrm{sir}. \\ $$

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