Question

in the diagram below, \\( \overline{ad} \parallel \overline{eg} \\), \\( \overline{bh} \perp \overline{fc} \\), and \\( m\angle gfh = 59^\circ \\). find \\( m\angle hbc \\).

Explanation:

Step1: Identify corresponding angles

Since \( \overline{AD} \parallel \overline{EG} \), the alternate interior angles \( \angle GFH \) and \( \angle BCF \) are equal. So \( m\angle BCF = m\angle GFH = 59^\circ \).

Step2: Use right triangle angle sum

In right triangle \( BHC \) (since \( \overline{BH} \perp \overline{FC} \), \( \angle BHC = 90^\circ \)), the sum of angles in a triangle is \( 180^\circ \). Let \( m\angle HBC = x \). Then \( x + 59^\circ + 90^\circ = 180^\circ \).

Step3: Solve for \( x \)

\( x = 180^\circ - 90^\circ - 59^\circ = 31^\circ \).

Answer:

\( 31^\circ \)