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Question Number 186594 by mr W last updated on 06/Feb/23

Commented by mr W last updated on 06/Feb/23

$$oldquestionrepostedforother$$$$solutions$$

Answered by CElcedricjunior last updated on 06/Feb/23

$$\frac{\mathit{c}}{\mathit{s}\mathit{i}\mathit{n}\mathit{\alpha }}=\frac{\mathit{a}+\mathit{b}}{\mathit{s}\mathit{i}\mathit{n}\left(110\right)}\mathit{o}\mathit{r}\frac{\mathit{b}}{\mathit{s}\mathit{i}\mathit{n}70}=\frac{\mathit{c}}{\mathit{s}\mathit{i}\mathit{n}80}$$$$=>\mathit{c}=\frac{\mathit{b}\mathit{s}\mathit{i}\mathit{n}80}{\mathit{s}\mathit{i}\mathit{n}70}$$$$<=>\frac{\mathit{b}\mathit{s}\mathit{i}\mathit{n}80}{\mathit{s}\mathit{i}\mathit{n}\mathit{\alpha }\mathit{s}\mathit{i}\mathit{n}70}=\frac{\mathit{a}+\mathit{b}}{\mathit{s}\mathit{i}\mathit{n}110}★\mathit{m}\mathit{o}\mathit{i}\mathit{v}\mathit{r}\mathit{e}$$$$=>\mathit{\alpha }={\mathit{s}\mathit{i}\mathit{n}}^{-1}\left[\frac{\mathit{b}}{(\mathit{a}+\mathit{b})}\frac{\mathit{s}\mathit{i}\mathit{n}\left(80\right)\times \mathit{s}\mathit{i}\mathit{n}\left(110\right)}{\mathit{s}\mathit{i}\mathit{n}\left(70\right)}\right]◼\mathit{c}\mathit{e}\mathit{d}\mathit{r}\mathit{i}\mathit{c}\mathit{j}\mathit{u}\mathit{n}\mathit{i}\mathit{o}\mathit{r}$$$$$$

Answered by HeferH last updated on 06/Feb/23

Commented by HeferH last updated on 06/Feb/23

$$70°>\alpha \Rightarrow$$$$\alpha \ne 140°\Rightarrow$$$$\alpha =40°$$

Answered by ajfour last updated on 06/Feb/23

$$let\alpha +\beta =70°,a=1$$$$\frac{c}{\sin 80°}=\frac{b}{\sin 70°}=\frac{1}{\sin 30°}=2$$$$\frac{b}{\sin \alpha }=\frac{1+b}{\sin 80°}$$$$\Rightarrow \sin \alpha =\frac{\sin 80°}{1+\frac{1}{2\sin 70°}}$$$$=\frac{2\sin 80°\sin 70°}{2\sin 70°+1}$$$$$$

Answered by Frix last updated on 06/Feb/23

$$\frac{a+b}{\sin 80}=\frac{b}{\sin \alpha }\wedge \frac{b}{\sin 70}=\frac{a}{\sin 30}$$$$\mathrm{Solve}\mathrm{both}\mathrm{for}a\mathrm{or}b\Rightarrow \mathrm{insert}\Rightarrow$$$$\alpha =40°$$

Answered by mr W last updated on 07/Feb/23

Commented by mr W last updated on 07/Feb/23

$$makeBD=BE=a$$$$AD=a+b=AC$$$$\mathrm{\angle }AED=\mathrm{\angle }AEC=110°$$$$\mathrm{\Delta }AED\equiv \mathrm{\Delta }AEC$$$$\alpha =\mathrm{\angle }ADE=40°$$

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