Question

parallel lines m and n are cut by transversal t. which statement is always true?
∠1 and ∠6 are complementary, and ∠3 and ∠7 are congruent.
∠1 and ∠6 are complementary, and ∠3 and ∠7 supplementary.
∠1 and ∠6 are supplementary, and ∠3 and ∠7 are complementary
∠1 and ∠6 are supplementary, and ∠3 and ∠7 are congruent.

Explanation:

Step1: Recall parallel - line angle relationships

When two parallel lines \(m\) and \(n\) are cut by a transversal \(t\), corresponding angles are congruent, alternate - interior angles are congruent, and same - side interior angles are supplementary.

Step2: Analyze \(\angle1\) and \(\angle6\)

\(\angle1\) and \(\angle6\) are same - side interior angles. By the property of parallel lines cut by a transversal, same - side interior angles are supplementary. So \(\angle1+\angle6 = 180^{\circ}\).

Step3: Analyze \(\angle3\) and \(\angle7\)

\(\angle3\) and \(\angle7\) are corresponding angles. Corresponding angles formed by two parallel lines cut by a transversal are congruent. So \(\angle3=\angle7\).

Answer:

\(\angle1\) and \(\angle6\) are supplementary, and \(\angle3\) and \(\angle7\) are congruent.