Question

look at this diagram:
if ( overleftrightarrow{df} ) and ( overleftrightarrow{gi} ) are parallel lines and ( mangle deh = 128^circ ), what is ( mangle ghj )?

Explanation:

Step1: Identify relationship (parallel lines, transversal)

Lines \( DF \) and \( GI \) are parallel, and \( JC \) is a transversal. \( \angle DEH \) and \( \angle GHJ \) are same - side interior angles? Wait, no, actually, \( \angle DEH \) and \( \angle GHE \) would be... Wait, no, let's correct. Since \( DF \parallel GI \), and \( JC \) is a transversal, \( \angle DEH \) and \( \angle GHJ \) are same - side interior angles? Wait, no, \( \angle DEH \) and \( \angle GHE \) are same - side interior angles? Wait, no, the sum of same - side interior angles is \( 180^{\circ} \) when lines are parallel. Wait, \( \angle DEH = 128^{\circ} \), and we need to find \( \angle GHJ \). Wait, actually, \( \angle DEH \) and \( \angle GHJ \) are same - side interior angles? Wait, no, let's think again. The lines \( DF \) and \( GI \) are parallel, cut by transversal \( JC \). So \( \angle DEH \) and \( \angle GHJ \) are same - side interior angles, so they are supplementary. So \( m\angle GHJ=180^{\circ}-m\angle DEH \)

Step2: Calculate \( m\angle GHJ \)

Given \( m\angle DEH = 128^{\circ} \), then \( m\angle GHJ=180 - 128=52^{\circ} \)

Answer:

\( 52^{\circ} \)