Question

with respect to the diagram, which relationship is false if ∠fea is supplementary to ∠hgd?
○ m∠feb + m∠hgd = 180°
○ ∠hgd ≅ ∠feb
○ m∠hgc + m∠feb = 180°
○ ∠fea ≅ ∠hgc

Explanation:

Step1: Recall supplementary - angle property

If \(\angle FEA\) is supplementary to \(\angle HGD\), then \(m\angle FEA + m\angle HGD=180^{\circ}\). Also, \(\angle FEA\) and \(\angle FEB\) are a linear - pair, so \(m\angle FEA + m\angle FEB = 180^{\circ}\).

Step2: Use substitution

From \(m\angle FEA + m\angle HGD=180^{\circ}\) and \(m\angle FEA + m\angle FEB = 180^{\circ}\), we can conclude that \(m\angle HGD=m\angle FEB\) (by the transitive property of equality), so \(\angle HGD\cong\angle FEB\) and \(m\angle FEB + m\angle HGD = 180^{\circ}\) is false.

Step3: Analyze vertical - angle and supplementary - angle relationships

\(\angle HGC\) and \(\angle HGD\) are a linear - pair (\(m\angle HGC + m\angle HGD = 180^{\circ}\)). Since \(m\angle HGD=m\angle FEB\), then \(m\angle HGC + m\angle FEB = 180^{\circ}\). Also, \(\angle FEA\) and \(\angle HGC\) are corresponding angles (assuming the lines are parallel based on the angle - relationship context), so \(\angle FEA\cong\angle HGC\).

Answer:

\(m\angle FEB + m\angle HGD = 180^{\circ}\)