ГДЗ по геометрии 7 класс Мерзляк Задание 107

\[Схематический\ рисунок.\]

\[Дано:\]

\[\angle ABF = 80{^\circ};\]

\[\angle ABD = 30{^\circ};\]

\[BM - биссектриса\ \angle DBF;\]

\[BN - биссектриса\ \angle FBC.\]

\[Найти:\]

\[\angle MBN.\]

\[Решение.\]

\[1)\ \angle ABF = \angle ABD + \angle DBF\]

\[80{^\circ} = 30{^\circ} + \angle DBF\]

\[\angle DBF = 50{^\circ}.\]

\[2)\ \angle DBF = \angle DBM + \angle MBF\]

\[\angle DBF = \angle MBF + \angle MBF\]

\[2\angle MBF = 50{^\circ}\ \ \ \]

\[\angle MBF = 25{^\circ}.\]

\[3)\ \angle ABF\ и\ \angle FBC\ смежные:\]

\[\angle ABF + \angle FBC = 180{^\circ}\]

\[80{^\circ} + \angle FBC = 180{^\circ}\]

\[\angle FBC = 100{^\circ}.\]

\[4)\ \angle FBC = \angle FBN + \angle NBC\]

\[\angle FBC = \angle FBN + \angle FBN\]

\[2\angle FBN = 100{^\circ}\ \ \ \]

\[\angle FBN = 50{^\circ}.\]

\[5)\ \angle MBN = \angle MBF + \angle FBN\]

\[\angle MBN = 25{^\circ} + 50{^\circ} = 75{^\circ}.\]

\[Ответ:\ \ 75{^\circ}.\]