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Question Number 193239 by Mingma last updated on 08/Jun/23

Answered by a.lgnaoui last updated on 08/Jun/23

$$\measuredangle \mathrm{B}=180-110=70$$$$△\mathrm{ABE}\measuredangle \mathrm{AEB}=180-120=60$$$$△\mathrm{ACE}\measuredangle \mathrm{AEC}=180-60=120$$$$$$$$△\mathrm{ABC}\frac{\sin 40}{\mathrm{AB}}=\frac{\sin 70}{\mathrm{BC}}$$$$△\mathrm{ACE}\mathrm{et}\mathrm{ABE}$$$$\frac{\sin 40}{\mathrm{AE}}=\frac{\sin 20}{\mathrm{CE}}\frac{\sin 70}{\mathrm{AE}}=\frac{\sin 50}{\mathrm{BE}}=\frac{\sin 60}{\mathrm{DB}}$$$$\Rightarrow \frac{\mathrm{CEsin}40}{\sin 20}=\frac{\mathrm{BEsin}70}{\sin 50}\left(1\right)$$$$$$$$△\mathrm{ABD}\mathrm{et}△\mathrm{BDE}$$$$\frac{\sin 50}{\mathrm{DB}}=\frac{\sin \mathrm{B}1}{\mathrm{AD}}$$$$\frac{\sin \alpha }{\mathrm{BE}}=\frac{\sin 60}{\mathrm{DB}}=\frac{\sin (70-\mathrm{B}1)}{\mathrm{DE}}$$$$\mathit{\alpha }=50+\mathrm{B}1$$$$\mathrm{DEsin}(50+\mathrm{B}1)=\mathrm{BEsin}(70-\mathrm{B}1)$$$$\frac{\mathrm{DE}}{\mathrm{BE}}=\frac{\sin (70-\mathrm{B}1)}{\sin (50+\mathrm{B}1)}\left(2\right)$$$$$$$$\left(1\right)\frac{\mathrm{CE}}{\mathrm{BE}}=\frac{\sin 70}{\sin 50}\times \frac{\sin 20}{\sin 40}$$$$\mathrm{CE}=\mathrm{DE}\Rightarrow$$$$\left(1\right)\mathrm{et}\left(2\right)\Rightarrow \frac{\sin 70\sin 20}{\sin 50\sin 40}=\frac{\sin (70-\mathrm{B}1)}{\sin (50+\mathrm{B}1)}$$$$=\frac{\sin 70\cos \mathrm{B}1-\cos 70\sin \mathrm{B}1}{\sin 50\cos \mathrm{B}1+\cos 50\sin \mathrm{B}1}$$$$=\frac{(\sin 70-\cos 70\mathrm{tanB}1)}{(\sin 50+\cos 50\tan \mathrm{B}1)}$$$$\Rightarrow \sin 50\times \sin 70\sin 20+\cos 50\tan \mathrm{B}1\times \sin 70\sin 20=$$$$\sin 70(\sin 50\sin 40-\cos 70\tan \mathrm{B}1\times \sin 50\sin 40$$$$$$$$\tan \mathrm{B}1(\cos 50\times \sin 70\sin 20+$$$$\cos 70\sin 50\sin 40)=$$$$\sin 70\sin 50\sin 40-\sin 50\sin 70\sin 20$$$$$$$$\tan \mathbf{B}1=\frac{\sin 70\sin 50(\sin 40-\sin 20)}{\cos 70\cos 50(\tan 70\sin 20+\tan 50\sin 40}$$$$$$$$=\frac{\tan 70\tan 50(\sin 40-\sin 20)}{\tan 70\sin 20+\tan 50\sin 40}$$$$$$$$\tan \mathbf{B}1=2,14039\Rightarrow \mathbf{B}1=65°$$$$$$$$\mathbf{alors}\mathit{\alpha }=50+\mathbf{B}1=115°$$$$$$$$$$

Answered by universe last updated on 09/Jun/23

Commented by Mingma last updated on 09/Jun/23

Perfect ��

Commented by a.lgnaoui last updated on 11/Jun/23

$$\mathrm{les}\mathrm{angles}40\mathrm{ne}\mathrm{sont}\mathrm{pas}\mathrm{les}\mathrm{memes}!$$$$\mathrm{verifier}\mathrm{votre}\mathrm{procedure}.$$

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