Question

in the diagram below, \\( \overline { a c } \parallel \overline { d g } \\), \\( \overline { b e } \cong \overline { b f } \\) and \\( m \angle e b f = 42 ^ { \circ } \\). find \\( m \angle h b c \\).

Explanation:

Step1: Find the base angles of isosceles triangle \( \triangle BEF \)

Since \( \overline{BE}\cong\overline{BF} \), \( \triangle BEF \) is isosceles. The sum of angles in a triangle is \( 180^{\circ} \). Let \( \angle BEF=\angle BFE = x \). Then \( x + x+42^{\circ}=180^{\circ} \), \( 2x=180^{\circ}- 42^{\circ}=138^{\circ} \), \( x = 69^{\circ} \). So \( \angle BFE=69^{\circ} \).

Step2: Use the property of parallel lines

Because \( \overline{AC}\parallel\overline{DG} \), \( \angle HBC=\angle BFE \) (corresponding angles).

Answer:

\( 69^{\circ} \)