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Question-186143

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Question Number 186143 by Rupesh123 last updated on 01/Feb/23
Answered by HeferH last updated on 01/Feb/23
2x + 24° = 180° x = 78°
$$\mathrm{2}{x}\:+\:\mathrm{24}°\:=\:\mathrm{180}° \\ $$$${x}\:=\:\mathrm{78}° \\ $$
Commented by HeferH last updated on 01/Feb/23
Commented by HeferH last updated on 01/Feb/23
Answered by a.lgnaoui last updated on 01/Feb/23
△DFG GD=DF ⇒30+∡EGD=24+18=42 ∡EGD=12 △DGF (2×42)+∡GDF=180 ∡GDF=96⇒ x=∡GDE=96−x △DGE ((sin 12)/(DE))=((sin (96−x))/(DG)) (1) △DEF ((sin 24)/(DE))=((sin x)/(EF))=((sin (24+x))/(DF)) (2) (1)⇒DGsin 12=DEsin (96−x) (2)⇒DFsin 24=DEsin (24+x) (((1))/((2)))⇒ ((sin 12)/(sin 24))=((sin (96−x))/(sin (24+x))) sin 12sin(24+x)=sin24sin(96−x) sin (24+x)=2cos12sin (96−x) tan x=((2cos 12sin 96−sin 24)/(cos 24+2cos 12cos 96)) =((1,538841768587)/(0,709056926535))=2,1702654765783 X=65°
$$\bigtriangleup{DFG}\:\:\:\:\:\mathrm{GD}=\mathrm{DF}\:\Rightarrow\mathrm{30}+\measuredangle{EGD}=\mathrm{24}+\mathrm{18}=\mathrm{42} \\ $$$$\measuredangle{EGD}=\mathrm{12} \\ $$$$\bigtriangleup{DGF}\:\:\:\:\left(\mathrm{2}×\mathrm{42}\right)+\measuredangle{GDF}=\mathrm{180} \\ $$$$\measuredangle{GDF}=\mathrm{96}\Rightarrow\:\:\:\:\mathrm{x}=\measuredangle{GDE}=\mathrm{96}−\mathrm{x} \\ $$$$\bigtriangleup{DGE}\:\:\:\frac{\mathrm{sin}\:\mathrm{12}}{{DE}}=\frac{\mathrm{sin}\:\left(\mathrm{96}−\mathrm{x}\right)}{{DG}}\:\:\:\:\:\left(\mathrm{1}\right) \\ $$$$\bigtriangleup{DEF}\:\:\frac{\mathrm{sin}\:\mathrm{24}}{{DE}}=\frac{\mathrm{sin}\:{x}}{{EF}}=\frac{\mathrm{sin}\:\left(\mathrm{24}+\mathrm{x}\right)}{\mathrm{D}{F}}\:\left(\mathrm{2}\right) \\ $$$$\left(\mathrm{1}\right)\Rightarrow{DG}\mathrm{sin}\:\mathrm{12}={DE}\mathrm{sin}\:\left(\mathrm{96}−\mathrm{x}\right) \\ $$$$\left(\mathrm{2}\right)\Rightarrow{DF}\mathrm{sin}\:\mathrm{24}={DE}\mathrm{sin}\:\left(\mathrm{24}+\mathrm{x}\right) \\ $$$$ \\ $$$$\frac{\left(\mathrm{1}\right)}{\left(\mathrm{2}\right)}\Rightarrow\:\frac{\mathrm{sin}\:\mathrm{12}}{\mathrm{sin}\:\mathrm{24}}=\frac{\mathrm{sin}\:\left(\mathrm{96}−\mathrm{x}\right)}{\mathrm{sin}\:\left(\mathrm{24}+\mathrm{x}\right)} \\ $$$$\mathrm{sin}\:\mathrm{12sin}\left(\mathrm{24}+\mathrm{x}\right)=\mathrm{sin24sin}\left(\mathrm{96}−\mathrm{x}\right) \\ $$$$\mathrm{sin}\:\left(\mathrm{24}+\mathrm{x}\right)=\mathrm{2cos12sin}\:\left(\mathrm{96}−\mathrm{x}\right) \\ $$$$\:\:\: \\ $$$$\mathrm{tan}\:\mathrm{x}=\frac{\mathrm{2cos}\:\mathrm{12sin}\:\mathrm{96}−\mathrm{sin}\:\mathrm{24}}{\mathrm{cos}\:\mathrm{24}+\mathrm{2cos}\:\mathrm{12cos}\:\mathrm{96}} \\ $$$$=\frac{\mathrm{1},\mathrm{538841768587}}{\mathrm{0},\mathrm{709056926535}}=\mathrm{2},\mathrm{1702654765783} \\ $$$$\:\:\:\:\:\:\:\boldsymbol{\mathrm{X}}=\mathrm{65}° \\ $$$$ \\ $$

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