Question

look at this diagram: if $overleftrightarrow{df}$ and $overleftrightarrow{gi}$ are parallel lines and $mangle fec = 44^circ$, what is $mangle ihe$? save answer

Explanation:

Step1: Identify the relationship

Since \( \overleftrightarrow{DF} \parallel \overleftrightarrow{GI} \) and \( \overleftrightarrow{EC} \) (or \( \overleftrightarrow{EJ} \)) is a transversal, \( \angle FEC \) and \( \angle IHE \) are same - side interior angles? Wait, no, actually, let's check the positions. Wait, \( \angle FEC \) and \( \angle IHE \): Wait, maybe they are supplementary? Wait, no, wait. Wait, \( \overleftrightarrow{DF} \parallel \overleftrightarrow{GI} \), and the transversal is \( \overleftrightarrow{EH} \) (or the line containing \( E \) and \( H \)). Wait, \( \angle FEC \) and \( \angle IHE \): Let's see, \( \angle FEC \) is given as \( 44^{\circ} \). Wait, actually, \( \angle FEC \) and \( \angle IHE \) are same - side interior angles? No, wait, maybe they are supplementary? Wait, no, wait. Wait, if \( DF \parallel GI \), then the consecutive interior angles should be supplementary. Wait, \( \angle FEC \) and \( \angle IHE \): Let's assume that the line \( EC \) (or \( EJ \)) is the transversal. So \( \angle FEC \) and \( \angle IHE \) are same - side interior angles, so they should be supplementary? Wait, no, wait, maybe I made a mistake. Wait, no, let's re - examine. Wait, \( \angle FEC \) and \( \angle IHE \): Wait, \( \angle FEC \) is at \( E \), between \( FE \) and \( EC \), and \( \angle IHE \) is at \( H \), between \( IH \) and \( HE \). Since \( DF \parallel GI \), then \( FE \parallel IH \) (because \( DF \) and \( GI \) are parallel, and \( FE \) is part of \( DF \), \( IH \) is part of \( GI \)). So the transversal is \( EH \). Then \( \angle FEC \) and \( \angle IHE \) are same - side interior angles, so \( m\angle FEC + m\angle IHE=180^{\circ} \)? Wait, no, that can't be. Wait, maybe \( \angle FEC \) and \( \angle IHE \) are alternate interior angles? No, alternate interior angles are equal. Wait, maybe I got the angles wrong. Wait, the problem says \( m\angle FEC = 44^{\circ} \), find \( m\angle IHE \). Wait, maybe \( \angle FEC \) and \( \angle IHE \) are supplementary. Wait, let's calculate: \( 180 - 44=136 \)? Wait, no, that doesn't seem right. Wait, maybe I misidentified the angles. Wait, maybe \( \angle FEC \) and \( \angle IHE \) are same - side interior angles. Wait, let's think again. If \( DF \parallel GI \), then the sum of same - side interior angles is \( 180^{\circ} \). So if \( \angle FEC = 44^{\circ} \), then \( \angle IHE=180 - 44 = 136^{\circ} \)? Wait, no, maybe I made a mistake. Wait, no, let's check the diagram again (mentally). The lines \( DF \) and \( GI \) are parallel. The transversal is the line that intersects both, say \( EH \) (the line from \( E \) to \( H \)). Then \( \angle FEC \) is on \( DF \) side, and \( \angle IHE \) is on \( GI \) side. So they are same - side interior angles, so they should be supplementary. So \( m\angle IHE = 180^{\circ}-m\angle FEC \).

Step2: Calculate the measure

Given \( m\angle FEC = 44^{\circ} \), then \( m\angle IHE=180^{\circ}- 44^{\circ}=136^{\circ} \). Wait, but that seems like a large angle. Wait, maybe I got the angle relationship wrong. Wait, maybe \( \angle FEC \) and \( \angle IHE \) are corresponding angles? No, corresponding angles are equal. Wait, maybe the angle is \( \angle FEC \) and \( \angle GHE \) are corresponding, but no. Wait, maybe I made a mistake in the angle identification. Wait, let's start over. The problem says \( \overleftrightarrow{DF} \parallel \overleftrightarrow{GI} \), \( m\angle FEC = 44^{\circ} \), find \( m\angle IHE \). Let's consider the transversal as \( \overleftrightarrow{EC} \) (or the line \( EJ \)). Then \( \angle FEC \) and \( \angle IHE \): If \( DF \parallel GI \), then \( \angle FEC \) and \( \angle IHE \) are same - side interior angles, so their sum is \( 180^{\circ} \). So \( m\angle IHE = 180 - 44=136 \).

Answer:

\( 136 \)