Question

look at this diagram: diagram with points e, b, a, c, f, h, g, d if bd and eg are parallel lines and m∠gfh= 58°, what is m∠dcf?

Explanation:

Step1: Identify angle relationship

Since \( BD \parallel EG \) and \( HA \) is a transversal, \( \angle GFH \) and \( \angle DCF \) are same - side interior angles? Wait, no. Wait, actually, \( \angle GFH \) and \( \angle DCF \): Let's re - examine. Wait, \( \angle GFH \) and \( \angle DCF \): Since \( BD\parallel EG \), and the transversal is \( HFAC \). Wait, \( \angle GFH \) and \( \angle DCF \) are same - side interior angles? No, wait, same - side interior angles add up to \( 180^{\circ} \). Wait, \( \angle GFH \) and \( \angle DCF \): Let's see, \( \angle GFH \) is at \( F \) between \( EG \) and \( HA \), and \( \angle DCF \) is at \( C \) between \( BD \) and \( HA \). Since \( BD\parallel EG \), consecutive interior angles (same - side interior angles) are supplementary. So \( m\angle GFH + m\angle DCF=180^{\circ} \).

Step2: Calculate \( m\angle DCF \)

We know that \( m\angle GFH = 58^{\circ} \). Using the formula for supplementary angles \( m\angle DCF=180^{\circ}-m\angle GFH \).
Substitute \( m\angle GFH = 58^{\circ} \) into the formula: \( m\angle DCF = 180^{\circ}- 58^{\circ}=122^{\circ} \).

Answer:

\( 122^{\circ} \)