Question

1. if ℓ || m, and m || n, how do we know ∠1 ≅ ∠12?

Explanation:

Step1: Transitive Property of Parallel Lines

Given \( \ell \parallel m \) and \( m \parallel n \), by the transitive property of parallel lines, we know \( \ell \parallel n \).

Step2: Corresponding Angles Postulate

When two parallel lines (\( \ell \) and \( n \)) are cut by a transversal, corresponding angles are congruent. \( \angle 1 \) and \( \angle 12 \) are corresponding angles formed by the transversal intersecting the parallel lines \( \ell \) and \( n \). So, by the Corresponding Angles Postulate, \( \angle 1 \cong \angle 12 \).

Answer:

First, by the transitive property of parallel lines, since \( \ell \parallel m \) and \( m \parallel n \), we get \( \ell \parallel n \). Then, by the Corresponding Angles Postulate (as \( \ell \parallel n \) and cut by a transversal), \( \angle 1 \cong \angle 12 \).