Question

in the diagram below, $overline{ad}paralleloverline{eg}$, find $mangle gfc$. step 1: $mangle bfc = 70^{circ}$, $mangle fbc = 37^{circ}$

Explanation:

Step1: Recall angle - sum property of a triangle

The sum of the interior angles of a triangle is 180°. In \(\triangle BFC\), we know two of the angles.

Step2: Calculate \(m\angle GFC\)

Let's assume we want to find \(m\angle GFC\) using the fact that in \(\triangle BFC\), if we know \(m\angle BFC = 70^{\circ}\) and \(m\angle FBC=37^{\circ}\), then by the angle - sum property of a triangle (\(m\angle BFC + m\angle FBC+m\angle BCF = 180^{\circ}\)). First, find \(m\angle BCF=180^{\circ}-(70^{\circ} + 37^{\circ})=73^{\circ}\). However, if we consider the linear - pair or other angle - relationships related to \(\angle GFC\), we note that if we assume the relevant angle relationship in the context of parallel lines \(AD\parallel EG\) (not fully utilized in the basic triangle calculation here). If we want to find \(m\angle GFC\) directly from the angles in \(\triangle BFC\), we know that \(\angle GFC\) and \(\angle BFC\) are supplementary (assuming a straight - line formation). So \(m\angle GFC=180^{\circ}-m\angle BFC\).
Since \(m\angle BFC = 70^{\circ}\), then \(m\angle GFC = 110^{\circ}\)

Answer:

\(110^{\circ}\)