Question

1. if $\angle 1$ and $\angle 7$ in the diagram are congruent, which theorem supports this relationship when lines $c$ and $b$ are parallel?

\bigcirc\quad alternate exterior angles theorem
\bigcirc\quad alternate interior angle theorem
\bigcirc\quad linear pair postulate
\bigcirc\quad corresponding angle theorem

Explanation:

To determine the theorem supporting the congruence of \( \angle 1 \) and \( \angle 7 \) when lines \( c \) and \( b \) are parallel, we analyze the angle positions:

• \( \angle 1 \) is an exterior angle above line \( b \), and \( \angle 7 \) is an exterior angle below line \( c \).
• These angles lie on opposite sides of the transversal (line \( a \)) and outside the two parallel lines (\( b \) and \( c \)).

The Alternate Exterior Angles Theorem states that if two parallel lines are cut by a transversal, then alternate exterior angles are congruent. This matches the relationship between \( \angle 1 \) and \( \angle 7 \).

Other options are incorrect:

• Alternate Interior Angles Theorem applies to angles between the parallel lines (e.g., \( \angle 4 \) and \( \angle 6 \)).
• Linear Pair Postulate applies to adjacent supplementary angles (e.g., \( \angle 1 \) and \( \angle 2 \)).
• Corresponding Angles Theorem applies to angles in the same relative position (e.g., \( \angle 1 \) and \( \angle 5 \)).

Answer:

Alternate Exterior Angles Theorem