Question

line l and m are cut by transversal n. which statement proves that line l is parallel to line m? 2nd attempt 6/12 options: m∠3 + m∠5 = 90°, m∠7 + m∠8 = 180°, ∠2 ≅ ∠6, ∠2 ≅ ∠3

Explanation:

To determine which statement proves line \( l \) is parallel to line \( m \), we analyze each option using angle - related parallel line theorems:

Step 1: Analyze Option 1 (\( m\angle3 + m\angle5 = 90^\circ \))

There is no standard parallel line theorem that states the sum of two angles (in the context of a transversal cutting two lines) being \( 90^\circ \) implies the lines are parallel. So this option does not prove \( l\parallel m \).

Step 2: Analyze Option 2 (\( m\angle7 + m\angle8 = 180^\circ \))

\( \angle7 \) and \( \angle8 \) are adjacent angles forming a linear pair. By the definition of a linear pair, their sum is always \( 180^\circ \), regardless of whether \( l \) and \( m \) are parallel. So this does not prove \( l\parallel m \).

Step 3: Analyze Option 3 (\( \angle2\cong\angle6 \))

\( \angle2 \) and \( \angle6 \) are corresponding angles. According to the Corresponding Angles Postulate, if two lines are cut by a transversal and the corresponding angles are congruent, then the two lines are parallel. So if \( \angle2\cong\angle6 \), we can conclude that \( l\parallel m \) by the Corresponding Angles Postulate.

Step 4: Analyze Option 4 (\( \angle2\cong\angle3 \))

\( \angle2 \) and \( \angle3 \) are adjacent angles (vertical angles or part of a linear pair - related angles). There is no parallel line theorem that uses the congruence of these two angles to prove two lines are parallel. So this option does not prove \( l\parallel m \).

Answer:

The statement that proves line \( l \) is parallel to line \( m \) is \( \boldsymbol{\angle2\cong\angle6} \) (the option with the orange background).