Question

problems 7-8
two parallel lines are crossed by a transversal.
write a conjecture about angles v and w.
describe a transformation that proves your conjecture. explain
your thinking.

Explanation:

Step1: Conjecture formation

Based on the property of parallel lines cut by a transversal, \(v\) and \(w\) are congruent. So the conjecture is \(\angle v\cong\angle w\).

Step2: Transformation for proof

A translation can be used. Translate the lower - parallel line (where \(\angle v\) is located) along the direction perpendicular to the parallel lines until it coincides with the upper - parallel line. Since translation is a rigid transformation (preserves shape and size), \(\angle v\) will map onto \(\angle w\).

Answer:

Conjecture: \(\angle v\cong\angle w\). Transformation: Translation. When we translate the line with \(\angle v\) (lower parallel line) along the perpendicular direction to the parallel lines to overlap with the line of \(\angle w\) (upper parallel line), because translation is a rigid motion (it doesn't change the measure of angles), \(\angle v\) is mapped to \(\angle w\), proving \(\angle v\cong\angle w\).