Question

look at this diagram:
if (overleftrightarrow{jl}) and (overleftrightarrow{mo}) are parallel lines and (mangle mnk = 68^{circ}), what is (mangle jkp)?
subrnit

Explanation:

Step1: Identify the relationship between angles

Since \( \overleftrightarrow{JL} \) and \( \overleftrightarrow{MO} \) are parallel, and \( \overleftrightarrow{IK} \) is a transversal, \( \angle WMK \) and \( \angle JKI \) are same - side interior angles? Wait, no, actually, when two parallel lines are cut by a transversal, same - side interior angles are supplementary, but wait, let's check the positions. Wait, \( \angle WMK \) and \( \angle JKI \): Wait, maybe they are same - side interior angles? Wait, no, let's re - examine. Wait, \( \overleftrightarrow{JL}\parallel\overleftrightarrow{MO} \), and the transversal is \( \overleftrightarrow{IP} \) (the line with points \( I, K, M, P \)). So \( \angle WMK \) and \( \angle JKI \): Wait, actually, \( \angle WMK \) and \( \angle JKI \) are same - side interior angles? Wait, no, same - side interior angles add up to \( 180^{\circ} \)? Wait, no, if the lines are parallel, same - side interior angles are supplementary. Wait, but let's check the diagram again. Wait, \( \angle WMK \) and \( \angle JKI \): Wait, maybe I made a mistake. Wait, actually, \( \angle WMK \) and \( \angle JKI \) are same - side interior angles? Wait, no, let's think about the parallel lines \( JL \) and \( MO \), cut by transversal \( IKMP \). So \( \angle JKI \) and \( \angle WMK \): if we consider the direction, \( \angle JKI \) and \( \angle WMK \) are same - side interior angles, so they should be supplementary. Wait, but the measure of \( \angle WMK = 68^{\circ} \), so \( m\angle JKI=180 - 68=112^{\circ} \)? Wait, no, maybe I got the angle relationship wrong. Wait, no, maybe they are alternate interior angles? Wait, no, alternate interior angles are equal. Wait, let's look at the positions. Point \( K \) is on \( JL \), point \( M \) is on \( MO \). The transversal is \( IKMP \). So \( \angle JKI \) and \( \angle OMP \) (but \( \angle WMK \) is at \( M \)). Wait, maybe \( \angle WMK \) and \( \angle JKI \) are same - side interior angles. So the sum of same - side interior angles is \( 180^{\circ} \) when lines are parallel. So \( m\angle JKI + m\angle WMK=180^{\circ} \).

Step2: Calculate \( m\angle JKI \)

We know that \( m\angle WMK = 68^{\circ} \), and since \( \overleftrightarrow{JL}\parallel\overleftrightarrow{MO} \), \( \angle JKI \) and \( \angle WMK \) are same - side interior angles, so they are supplementary. That is, \( m\angle JKI + m\angle WMK=180^{\circ} \).
Substitute \( m\angle WMK = 68^{\circ} \) into the equation:
\( m\angle JKI=180^{\circ}- 68^{\circ}=112^{\circ} \)

Answer:

\( 112 \)