Question

look at this diagram: if $\overleftrightarrow{df}$ and $\overleftrightarrow{gi}$ are parallel lines and $m\angle feh = 53^\circ$, what is $m\angle ihj$?

Explanation:

Step1: Identify Angle Relationship

Since \( \overleftrightarrow{DF} \parallel \overleftrightarrow{GI} \) and \( \overleftrightarrow{CJ} \) is a transversal, \( \angle FEH \) and \( \angle IHJ \) are same - side interior angles? Wait, no, actually, \( \angle FEH \) and \( \angle IHJ \): Wait, let's re - examine. Wait, \( \angle FEH \) and \( \angle IHJ \): Wait, \( \overleftrightarrow{DF}\parallel\overleftrightarrow{GI} \), and the transversal is \( \overleftrightarrow{CJ} \). \( \angle FEH \) and \( \angle IHJ \): Wait, actually, \( \angle FEH \) and \( \angle IHJ \) are same - side interior angles? No, wait, \( \angle FEH \) and \( \angle IHJ \): Wait, no, \( \angle FEH \) and \( \angle GHE \) would be... Wait, no, the correct relationship: Since \( DF\parallel GI \), and the transversal is \( CJ \), \( \angle FEH \) and \( \angle IHJ \) are same - side interior angles? Wait, no, actually, \( \angle FEH \) and \( \angle IHJ \) are supplementary? Wait, no, wait, maybe I made a mistake. Wait, \( \angle FEH \) and \( \angle IHJ \): Wait, \( DF\parallel GI \), so the consecutive interior angles (same - side interior angles) are supplementary. Wait, \( \angle FEH \) and \( \angle EHI \): Wait, maybe the angle we need is \( \angle IHJ \), but wait, the problem says \( m\angle FEH = 53^{\circ} \), and we need to find \( m\angle IHJ \). Wait, no, actually, \( \angle FEH \) and \( \angle IHJ \) are same - side interior angles? Wait, no, let's think again. If two parallel lines are cut by a transversal, same - side interior angles are supplementary. Wait, \( DF\parallel GI \), transversal \( CJ \). So \( \angle FEH \) and \( \angle EHI \) are same - side interior angles? Wait, no, \( E \) is on \( DF \), \( H \) is on \( GI \). So \( \angle FEH \) and \( \angle IHJ \): Wait, maybe \( \angle FEH \) and \( \angle IHJ \) are supplementary? Wait, no, wait, \( \angle FEH \) is \( 53^{\circ} \), and if we consider that \( \angle FEH \) and \( \angle IHJ \) are same - side interior angles, then \( m\angle FEH + m\angle IHJ=180^{\circ} \)? Wait, no, that can't be. Wait, maybe I got the angle wrong. Wait, the problem says "what is \( m\angle IHJ \)". Wait, maybe \( \angle FEH \) and \( \angle IHJ \) are supplementary? Wait, no, let's check the diagram again (mentally). \( DF \) and \( GI \) are parallel, transversal \( CJ \). \( E \) is the intersection of \( CJ \) and \( DF \), \( H \) is the intersection of \( CJ \) and \( GI \). So \( \angle FEH \) is at \( E \), between \( FE \) and \( EH \), and \( \angle IHJ \) is at \( H \), between \( IH \) and \( HJ \). So these two angles are same - side interior angles, so they should be supplementary. Wait, but \( m\angle FEH = 53^{\circ} \), so \( m\angle IHJ=180 - 53=127^{\circ} \)? Wait, no, that doesn't seem right. Wait, maybe they are alternate interior angles? Wait, no, alternate interior angles would be \( \angle FEH \) and \( \angle GHE \). Wait, maybe the problem has a typo, or I misread the angle. Wait, the problem says "what is \( m\angle IHN \)"? Wait, maybe it's \( \angle IHJ \) or \( \angle IHN \). Wait, maybe I made a mistake. Wait, let's start over.

Given \( DF\parallel GI \), transversal \( CJ \). \( \angle FEH = 53^{\circ} \). We need to find \( m\angle IHJ \) (assuming \( IHN \) is a typo for \( IHJ \)). Since \( DF\parallel GI \), and \( CJ \) is a transversal, \( \angle FEH \) and \( \angle IHJ \) are same - side interior angles. The sum of same - side interior angles is \( 180^{\circ} \).

Step2: Calculate the Angle

We know that for two parallel lines cut by a transversal, same - side interior angles are supplementary. So \( m\angle FEH + m\angle IHJ = 180^{\circ} \).

We are given that \( m\angle FEH = 53^{\circ} \).

So, \( m\angle IHJ=180^{\circ}-m\angle FEH \)

Substitute \( m\angle FEH = 53^{\circ} \) into the formula:

\( m\angle IHJ = 180 - 53=127^{\circ} \)

Wait, but maybe the angle is a corresponding angle? Wait, no, if \( DF\parallel GI \), and the transversal is \( CJ \), \( \angle FEH \) and \( \angle GHJ \) would be corresponding angles, but \( \angle IHJ \) is adjacent to \( \angle GHJ \) (they are supplementary). Wait, maybe the problem is that \( \angle FEH \) and \( \angle IHJ \) are supplementary. So the calculation is \( 180 - 53 = 127 \).

Answer:

\( 127 \)