Question

look at this diagram:
diagram of two parallel lines ik and ln cut by a transversal ho
if \\(\overleftrightarrow{ik}\\) and \\(\overleftrightarrow{ln}\\) are parallel lines and \\(m\angle lmo = 48.7^\circ\\), what is \\(m\angle nmj\\)?

Explanation:

Step1: Identify angle relationship

Since \( \overleftrightarrow{IK} \parallel \overleftrightarrow{LN} \) and \( \overleftrightarrow{HO} \) is a transversal, \( \angle LMO \) and \( \angle NMJ \) are same - side interior angles? Wait, no. Wait, \( \angle LMO \) and \( \angle NMJ \): Wait, actually, \( \angle LMO \) and \( \angle NMJ \) are supplementary? Wait, no. Wait, let's look at the lines. \( \overleftrightarrow{IK} \parallel \overleftrightarrow{LN} \), and the transversal is \( \overleftrightarrow{HO} \). \( \angle LMO \) and \( \angle NMJ \): Wait, \( \angle LMO \) and \( \angle NMJ \) are same - side interior angles? No, wait, \( \angle LMO \) and \( \angle NMJ \): Wait, actually, \( \angle LMO \) and \( \angle NMJ \) are supplementary because they form a linear pair? No, wait, no. Wait, \( \angle LMO \) and \( \angle NMJ \): Let's see, \( \angle LMO \) and \( \angle NMJ \): Wait, \( \overleftrightarrow{LN} \) is a straight line, so \( \angle LMO + \angle NMJ=180^{\circ} \)? Wait, no, that's not right. Wait, no, \( \angle LMO \) and \( \angle NMJ \): Wait, \( \overleftrightarrow{IK} \parallel \overleftrightarrow{LN} \), and the transversal is \( \overleftrightarrow{HO} \). Wait, \( \angle LMO \) and \( \angle KJM \) would be alternate interior angles, but here we have \( \angle NMJ \). Wait, no, let's re - examine. \( \angle LMO \) and \( \angle NMJ \): Since \( \overleftrightarrow{LN} \) is a straight line, \( \angle LMO + \angle OMN = 180^{\circ} \), but \( \angle OMN \) and \( \angle NMJ \): Wait, no, \( \overleftrightarrow{IK} \parallel \overleftrightarrow{LN} \), so \( \angle KJM=\angle LMO \) (alternate interior angles), and \( \angle KJM + \angle NMJ = 180^{\circ} \) (linear pair). So \( \angle LMO+\angle NMJ = 180^{\circ} \).

Step2: Calculate \( m\angle NMJ \)

We know that \( m\angle LMO = 48.7^{\circ} \). Since \( \angle LMO \) and \( \angle NMJ \) are supplementary (they form a linear pair with the transversal and parallel lines), we use the formula \( m\angle NMJ=180^{\circ}-m\angle LMO \).

Substitute \( m\angle LMO = 48.7^{\circ} \) into the formula: \( m\angle NMJ = 180^{\circ}- 48.7^{\circ}=131.3^{\circ} \).

Answer:

\( 131.3 \)