Question

question
in the diagram below, \\( \overline { b e } \cong \overline { b c } \\), \\( m \angle a = 40 ^ { \circ } \\) and \\( m \angle a d c = 105 ^ { \circ } \\). find \\( m \angle a f b \\).
you may assume lines that appear straight are straight, but the figure is not otherwise drawn to scale.

Explanation:

Step1: 利用全等三角形性质

因为\(\overline{BE}\cong\overline{BC}\)，\(\angle A = 40^{\circ}\)，\(\angle ADC=105^{\circ}\)。根据三角形外角性质，\(\angle AFB\)与\(\angle DFC\)是对顶角，\(\angle ADC\)是\(\triangle DFC\)外角，\(\angle A\)是\(\triangle ABF\)内角。

Step2: 计算\(\angle AFB\)

\(\angle AFB = 180^{\circ}-(180^{\circ}-\angle ADC)-\angle A\)，将\(\angle A = 40^{\circ}\)，\(\angle ADC = 105^{\circ}\)代入，\(\angle AFB=180^{\circ}-(180^{\circ} - 105^{\circ})-40^{\circ}=65^{\circ}\)。

Answer:

\(65^{\circ}\)