Question

$\angle 1\cong \angle 6$
a) $p \parallel q$; converse of corresponding angles theorem
b) $p \parallel q$; alternate interior angles converse
c) $g \parallel h$; converse of corresponding angles theorem
d) $g \parallel h$; alternate interior angles converse

Explanation:

1. First, identify the angles and lines: \( \angle 1 \) and \( \angle 6 \) are alternate interior angles formed by transversal \( p \) (or another transversal) with lines \( g \) and \( h \).
2. The Alternate Interior Angles Converse states that if alternate interior angles are congruent, then the two lines cut by the transversal are parallel.
3. Since \( \angle 1 \cong \angle 6 \), by the Alternate Interior Angles Converse, lines \( g \) and \( h \) must be parallel (\( g \parallel h \)).
4. Options A and B involve lines \( p \) and \( q \), but \( \angle 1 \) and \( \angle 6 \) are not related to \( p \parallel q \) (they are not alternate interior/corresponding angles for \( p \) and \( q \)). Option C refers to the Converse of Corresponding Angles Theorem, but \( \angle 1 \) and \( \angle 6 \) are alternate interior, not corresponding, angles. Thus, D is correct.

Answer:

D) \( g \parallel h \); Alternate Interior Angles Converse