Question

look at this diagram: diagram of lines jk, ln (parallel) cut by transversal hd with points j, m on transversal; points i, k on jk; l, n on ln; h, d on transversal if jk and ln are parallel lines and ( mangle ljh = 110^circ ), what is ( mangle nmd )?

Explanation:

Step1: Identify Angle Relationship

Since \( IK \parallel LN \) and \( HD \) is a transversal, \( \angle LJH \) and \( \angle NMD \) are corresponding angles? Wait, no, actually \( \angle LJH \) and \( \angle JMN \) would be corresponding, but wait, \( \angle LJH = 110^\circ \), and \( \angle NMD \) and \( \angle LJH \) – wait, actually, \( \angle LJH \) and \( \angle JMN \) are corresponding, but \( \angle NMD \) and \( \angle HJK \) might be vertical or supplementary? Wait, no, let's correct. Wait, \( IK \) and \( LN \) are parallel, cut by transversal \( HD \). So \( \angle LJH \) and \( \angle NMD \) – wait, \( \angle LJH \) is at \( J \), and \( \angle NMD \) is at \( M \). Wait, actually, \( \angle LJH \) and \( \angle JMN \) are corresponding angles (since \( IK \parallel LN \), same transversal), so \( \angle JMN = 110^\circ \). But \( \angle NMD \) and \( \angle JMN \) are supplementary? Wait, no, \( \angle JMN \) and \( \angle NMD \) – wait, \( HD \) is a straight line, so \( \angle JMN + \angle NMD = 180^\circ \)? Wait, no, maybe I misread the angle. Wait, the problem says \( m\angle LJH = 110^\circ \), and we need \( m\angle NMD \). Wait, actually, \( \angle LJH \) and \( \angle NMD \) are alternate interior angles? No, wait, \( IK \parallel LN \), transversal \( HD \). So \( \angle LJH \) and \( \angle NMD \) – wait, \( \angle LJH \) is at \( J \), between \( IK \) and \( HD \), and \( \angle NMD \) is at \( M \), between \( LN \) and \( HD \). Wait, actually, \( \angle LJH \) and \( \angle NMD \) are corresponding angles? No, maybe vertical angles or supplementary. Wait, no, let's think again. If \( IK \parallel LN \), then \( \angle LJH \) and \( \angle JMN \) are corresponding angles, so \( \angle JMN = 110^\circ \). Then \( \angle NMD \) and \( \angle JMN \) are supplementary because they form a linear pair? Wait, no, \( \angle JMN \) and \( \angle NMD \) – wait, \( HD \) is a straight line, so \( \angle JMH \) and \( \angle NMD \) are vertical angles? Wait, maybe I made a mistake. Wait, the correct approach: \( \angle LJH = 110^\circ \), and \( IK \parallel LN \), so \( \angle LJH \) and \( \angle NMD \) – wait, actually, \( \angle LJH \) and \( \angle NMD \) are equal? No, that can't be. Wait, no, \( \angle LJH \) and \( \angle JMN \) are corresponding angles (since \( IK \parallel LN \), transversal \( HD \)), so \( \angle JMN = 110^\circ \). Then \( \angle NMD \) and \( \angle JMN \) are supplementary because they are adjacent angles on a straight line (linear pair), so \( m\angle NMD = 180^\circ - 110^\circ = 70^\circ \). Wait, no, wait, maybe \( \angle LJH \) and \( \angle NMD \) are alternate interior angles? Wait, no, let's check the diagram again. The lines \( IK \) (with points \( I, J, K \)) and \( LN \) (with points \( L, M, N \)) are parallel, cut by transversal \( HD \) (with points \( H, J, M, D \)). So \( \angle LJH \) is at \( J \), between \( IK \) (going from \( I \) to \( K \)) and \( HD \) (going from \( H \) to \( D \)). \( \angle NMD \) is at \( M \), between \( LN \) (going from \( L \) to \( N \)) and \( HD \) (going from \( H \) to \( D \)). So \( \angle LJH \) and \( \angle NMD \) are actually corresponding angles? Wait, no, \( IK \) is horizontal (from \( I \) left to \( K \) right), \( LN \) is horizontal (from \( L \) left to \( N \) right), so they are parallel horizontal lines. Transversal \( HD \) is a diagonal line, going from \( H \) up to \( D \) down. So \( \angle LJH \) is the angle at \( J \) between \( IK \) (leftward) and \( HD \) (upward). \( \angle NMD \) is the angle at \( M \) between \( LN \) (rightward) and \( HD \) (downward). Wait, maybe \( \angle LJH \) and \( \angle NMD \) are vertical angles? No, vertical angles are opposite each other. Wait, \( \angle LJH \) and \( \angle KJH \) are supplementary (linear pair), but \( IK \) is straight, so \( \angle LJH + \angle KJH = 180^\circ \), so \( \angle KJH = 70^\circ \). Then, since \( IK \parallel LN \), \( \angle KJH \) and \( \angle NMD \) are corresponding angles (because \( IK \parallel LN \), transversal \( HD \)), so \( \angle NMD = \angle KJH = 70^\circ \). Ah, that makes sense. So step by step:

Step1: Find Supplementary Angle to \( \angle LJH \)

Since \( IK \) is a straight line, \( \angle LJH \) and \( \angle KJH \) are supplementary (linear pair). So:
\( m\angle KJH = 180^\circ - m\angle LJH \)
\( m\angle KJH = 180^\circ - 110^\circ = 70^\circ \)

Step2: Use Corresponding Angles (Parallel Lines)

Since \( IK \parallel LN \) and \( HD \) is a transversal, \( \angle KJH \) and \( \angle NMD \) are corresponding angles. Corresponding angles are equal when lines are parallel. So:
\( m\angle NMD = m\angle KJH = 70^\circ \)

Answer:

\( 70^\circ \)