Решебник по геометрии 7 класс Мерзляк Задание 108

\[Схематический\ рисунок.\]

\[Дано:\ \ \]

\[OD - биссектриса\ \angle AOB;\]

\[\angle BOD = \angle BOC - 18{^\circ}.\]

\[Найти:\]

\[\angle AOB;\ \angle BOC.\]

\[Решение.\]

\[1)\ \angle AOB = \angle AOD + \angle BOD\]

\[\angle AOB = \angle BOD + \angle BOD\]

\[\angle AOB = 2\angle BOD.\]

\[2)\ \angle AOB\ и\ \angle BOC\ смежные:\]

\[\angle AOB + \angle BOC = 180{^\circ}\]

\[2\angle BOD + \angle BOC = 180{^\circ}\]

\[2(\angle BOC - 18{^\circ}) + \angle BOC = 180{^\circ}\]

\[2\angle BOC - 36{^\circ} + \angle BOC = 180{^\circ}\]

\[3\angle BOC = 216{^\circ}\]

\[\angle BOC = 72{^\circ}.\]

\[\angle BOD = 72{^\circ} - 18{^\circ} = 54{^\circ}.\]

\[\angle AOB = 2 \bullet 54{^\circ} = 108{^\circ}.\]

\[Ответ:\ \ \angle AOB = 108{^\circ};\ \angle BOC = 72{^\circ}.\]