Question

in the diagram below, \\( \overline { a c } \parallel \overline { d g } \\), \\( \overline { b e } \cong \overline { b f } \\) and \\( m \angle a b h = 66 ^ { \circ } \\). find \\( m \angle a b e \\).

Explanation:

Step1: Find the measure of \(\angle EBF\)

Since \(\overline{AC}\parallel\overline{DG}\), \(\angle ABH\) and \(\angle BEF\) are corresponding angles. So \(\angle BEF=\angle ABH = 66^{\circ}\).
Because \(\overline{BE}\cong\overline{BF}\), \(\triangle BEF\) is isosceles. Let \(m\angle BFE=m\angle BEF = 66^{\circ}\).
Using the angle - sum property of a triangle (\(m\angle EBF+ m\angle BEF+m\angle BFE = 180^{\circ}\)), we have \(m\angle EBF=180-(66 + 66)=48^{\circ}\).

Step2: Find the measure of \(\angle ABE\)

\(\angle ABE+\angle EBF+\angle FBG = 180^{\circ}\) (linear - pair). Also, \(\angle FBG=\angle ABH = 66^{\circ}\) (vertical angles).
Let \(m\angle ABE=x\). Then \(x + 48+66=180\).
\(x=180-(48 + 66)=66^{\circ}\)

Answer:

\(66^{\circ}\)