Question

eg and hj are parallel lines. which angles are supplementary angles? ∠gfi and ∠gfd ∠gfi and ∠hif ∠gfi and ∠jik ∠gfi and ∠efd

Explanation:

Step1: Recall the definition of supplementary angles

Supplementary angles are two angles whose sum is \(180^{\circ}\).

Step2: Analyze \(\angle GFI\) and \(\angle GFD\)

Since \(\angle GFI\) and \(\angle GFD\) form a linear pair (they are adjacent and their non - common sides are opposite rays), by the linear pair postulate, \(\angle GFI+\angle GFD = 180^{\circ}\).

Step3: Analyze \(\angle GFI\) and \(\angle HIF\)

\(\overrightarrow{EG}\parallel\overrightarrow{HJ}\), but \(\angle GFI\) and \(\angle HIF\) are not in a relationship (such as same - side interior angles, linear pair) that would make their sum \(180^{\circ}\).

Step4: Analyze \(\angle GFI\) and \(\angle JIK\)

\(\angle JIK\) and \(\angle HIF\) are vertical angles (\(\angle JIK=\angle HIF\)). Since \(\overrightarrow{EG}\parallel\overrightarrow{HJ}\), \(\angle GFI\) and \(\angle HIF\) are not supplementary (they are alternate interior angles if we consider the transversal \(KI D\) cutting the parallel lines \(\overrightarrow{EG}\) and \(\overrightarrow{HJ}\), and alternate interior angles are equal when lines are parallel, not supplementary). So \(\angle GFI\) and \(\angle JIK\) are not supplementary.

Step5: Analyze \(\angle GFI\) and \(\angle EFD\)

\(\angle EFD\) and \(\angle GFD\) are vertical angles (\(\angle EFD=\angle GFD\)). \(\angle GFI\) and \(\angle EFD\) are not supplementary.

Answer:

\(\angle GFI\) and \(\angle GFD\)