Question

2. In figura alăturată, BD este mediatoarea segmentului AC, iar
AB= 2 OB. Măsura unghiului ABC este de:
rapid , va rooog​

Answer 1

$\bf BD - mediatoare \implies \begin{cases} \bf AO = CO (1) \\ \\ \bf \angle AOB = \angle BOC=90^{\circ} \end{cases}$

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$\left.\begin{aligned} \bf \angle AOB = 90^{\circ} \\ \\ \bf AB=2\, OB\; \; \; \end{aligned} \right\} \bf \implies \angle BAO = 30^{\circ}$

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$\bf \left.\begin{aligned}\bf BO \in \triangle AOB \; \\ \\ \bf BO \in \triangle BOC \; \end{aligned} \right\} \implies BO=BO \; \; (2)$

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$\left.\begin{aligned} \bf AO = CO\; \; \; \; \; \; \; \; \; \; \; \; \\\\ \bf \angle AOB=\angle COB =90^{\circ} \; \\\\\bf BO=BO \; \; \; \; \; \; \; \; \; \; \; \; \end{aligned} \right\} \xrightarrow{\;\;\;\bf C.C.\;\;\;} \underbrace{\bf \triangle AOB = \triangle COB }_\text{*}$

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$\bf \;$$\bf \triangle AOB=\triangle COB \implies \begin{cases} \bf AB=BC \\ \\ \bf \angle BAO = \angle BCO=30^{\circ} \end{cases}$

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$\left.\begin{aligned}\bf \angle BAO+\angle BCO + \angle ABC=180^{\circ}\\\\ \bf \angle BAO=\angle BCO = 30^{\circ} \;\;\;\;\; \;\; \end{aligned} \right\} \implies \boxed{\bf \angle ABC = 120^{\circ}}$

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