Question

look at this diagram:
diagram showing lines eg, hj, and transversal dk with points f, i
if \\( \overleftrightarrow{eg} \\) and \\( \overleftrightarrow{hj} \\) are parallel lines and \\( m\angle jif = 136^\circ \\), what is \\( m\angle gfi \\)?
\\( \square^\circ \\)

Explanation:

Step1: Identify Angle Relationship

Since \( \overleftrightarrow{EG} \parallel \overleftrightarrow{HJ} \) and \( \overleftrightarrow{DK} \) is a transversal, \( \angle JIF \) and \( \angle GFI \) are same - side interior angles? Wait, no. Wait, \( \angle JIF \) and \( \angle GFI \): Wait, actually, \( \angle JIF \) and \( \angle GFI \) are same - side interior angles? Wait, no, let's check the positions. \( \angle JIF \) and \( \angle GFI \): Wait, \( \angle JIF \) is at \( I \), and \( \angle GFI \) is at \( F \). Wait, actually, \( \angle JIF \) and \( \angle GFI \) are same - side interior angles? Wait, no, when two parallel lines are cut by a transversal, same - side interior angles are supplementary. Wait, \( \angle JIF \) and \( \angle GFI \): Let's see, \( \overleftrightarrow{EG} \parallel \overleftrightarrow{HJ} \), transversal \( DK \). So \( \angle JIF \) and \( \angle GFI \): Wait, \( \angle JIF \) is adjacent to \( \angle HIF \), and \( \angle HIF \) and \( \angle GFI \) are alternate interior angles? No, wait, \( \angle JIF = 136^{\circ}\), and \( \angle JIF \) and \( \angle HIF \) are supplementary (linear pair), so \( m\angle HIF=180 - 136 = 44^{\circ}\). Then, since \( \overleftrightarrow{EG}\parallel\overleftrightarrow{HJ}\), \( \angle GFI=\angle HIF \) (alternate interior angles). Wait, no, let's correct. \( \angle JIF \) and \( \angle GFI \): Wait, \( \angle JIF \) and \( \angle GFI \) are same - side interior angles? Wait, no, \( \angle JIF \) is on line \( HJ \), and \( \angle GFI \) is on line \( EG \). The transversal is \( DK \). So \( \angle JIF \) and \( \angle GFI \): Wait, actually, \( \angle JIF \) and \( \angle GFI \) are same - side interior angles, so they should be supplementary? Wait, no, that can't be. Wait, let's draw the lines. \( EG \) and \( HJ \) are parallel. Transversal \( DK \). At point \( I \), \( \angle JIF = 136^{\circ}\), so the adjacent angle \( \angle HIF=180 - 136 = 44^{\circ}\). Then, \( \angle HIF \) and \( \angle GFI \) are alternate interior angles, so \( \angle GFI=\angle HIF \). Wait, no, alternate interior angles are equal. Wait, \( \angle HIF \) and \( \angle GFI \): \( HJ \parallel EG \), transversal \( DK \), so \( \angle HIF \) (at \( I \), between \( HJ \) and \( DK \)) and \( \angle GFI \) (at \( F \), between \( EG \) and \( DK \)) are alternate interior angles, so they are equal. But \( \angle HIF = 180^{\circ}-\angle JIF=180 - 136 = 44^{\circ}\), so \( \angle GFI = 44^{\circ}\). Wait, let's verify: same - side interior angles: \( \angle JIF \) and \( \angle GFI \): if \( EG \parallel HJ \), then same - side interior angles are supplementary. Wait, \( \angle JIF + \angle GFI=180^{\circ}\)? But \( 136 + 44 = 180 \), yes! So \( \angle JIF \) and \( \angle GFI \) are same - side interior angles, so they are supplementary. So \( m\angle GFI = 180 - 136=44^{\circ}\).

Step2: Calculate the Measure of \( \angle GFI \)

Since \( \overleftrightarrow{EG}\parallel\overleftrightarrow{HJ} \) and \( \overleftrightarrow{DK} \) is a transversal, \( \angle JIF \) and \( \angle GFI \) are same - side interior angles. Same - side interior angles are supplementary, so \( m\angle GFI=180^{\circ}-m\angle JIF \).
Given \( m\angle JIF = 136^{\circ}\), then \( m\angle GFI=180 - 136 = 44^{\circ}\).

Answer:

\( 44 \)