Question

diagram: horizontal lines l (top) and m (bottom). transversals n (left, slanting upward) and p (right, slanting upward). angles: ∠1 (between n and l), ∠2 (between p and l), ∠3 (between n and m). four proof tables:

1. statements: ∠1 ≅ ∠2 (reason: alternate interior angles are congruent); ∠1 ≅ ∠3 (reason: corresponding angles are congruent); ∠2 ≅ ∠3 (reason: transitive property of congruence)

1. statements: ∠1 ≅ ∠2 (reason: corresponding angles are congruent); ∠1 ≅ ∠3 (reason: corresponding angles are congruent); ∠2 ≅ ∠3 (reason: transitive property of congruence)

1. statements: ∠1 ≅ ∠2 (reason: corresponding angles are congruent); ∠1 ≅ ∠3 (reason: alternate interior angles are congruent); ∠2 ≅ ∠3 (reason: transitive property of congruence)

1. statements: ∠1 ≅ ∠2 (reason: corresponding angles are congruent); ∠1 ≅ ∠3 (reason: corresponding angles are congruent); ∠2 ≅ ∠3 (reason: corresponding parts of congruent triangles are congruent) (text cut off)

Explanation:

To determine the correct proof, we analyze the angle relationships:

• ∠1 ≅ ∠2: These are corresponding angles (formed by transversal \( p \) with parallel lines \( l \) and \( m \)), so "Corresponding angles are congruent" is correct.
• ∠1 ≅ ∠3: These are alternate interior angles (formed by transversal \( n \) with parallel lines \( l \) and \( m \)), so "Alternate interior angles are congruent" is correct.
• ∠2 ≅ ∠3: By the Transitive Property of Congruence (if \( ∠1 ≅ ∠2 \) and \( ∠1 ≅ ∠3 \), then \( ∠2 ≅ ∠3 \)).

Looking at the options, the third table (from top) has:

• \( ∠1 ≅ ∠2 \): Corresponding angles are congruent.
• \( ∠1 ≅ ∠3 \): Alternate interior angles are congruent.
• \( ∠2 ≅ ∠3 \): Transitive Property of Congruence.

This matches the correct angle relationships.

Answer:

The third proof table (with \( \angle 1 \cong \angle 2 \) as corresponding angles, \( \angle 1 \cong \angle 3 \) as alternate interior angles, and \( \angle 2 \cong \angle 3 \) by transitive property).