Question

we know that angle 1 is congruent to angle 3 and that line a is parallel to line b because they are given. we see that ______ by the alternate exterior angles theorem. therefore, angle 2 is congruent to angle 3 by the transitive property. so, we can conclude that lines e and f are parallel by the converse alternate exterior angles theorem. which information is missing in the paragraph proof? options: ∠2 ≅ ∠4, ∠1 ≅ ∠2, ∠2 ≅ ∠3, ∠1 ≅ ∠4

Explanation:

The alternate exterior angles theorem states that if two parallel lines are cut by a transversal, then alternate exterior angles are congruent. Here, lines \(a\) and \(b\) are parallel (given), and the transversal would create alternate exterior angles. For the alternate exterior angles theorem to apply, \(\angle1\) and \(\angle2\) should be alternate exterior angles (since \(a\parallel b\) and cut by a transversal, so \(\angle1\cong\angle2\) by alternate exterior angles theorem). Then, using transitivity with \(\angle1\cong\angle3\) (given), we get \(\angle2\cong\angle3\). Let's analyze the options:

• Option \(\angle2\cong\angle4\): These are vertical angles, but not related to the alternate exterior angles theorem here.
• Option \(\angle1\cong\angle2\): This is what's needed for the alternate exterior angles theorem (since \(a\parallel b\), alternate exterior angles \(\angle1\) and \(\angle2\) are congruent).
• Option \(\angle2\cong\angle3\): This is the result of transitivity, not the step from the alternate exterior angles theorem.
• Option \(\angle1\cong\angle4\): These are not related by alternate exterior angles theorem.

Answer:

B. \(\boldsymbol{\angle1 \cong \angle2}\)