Question

13.
in the diagram, it is given that $\angle 2 \cong \angle 5$. lines $n$ and $p$ are parallel. which theorem best justifies why $\angle 2 \cong \angle 5$?
\bigcirc \quad vertical angles theorem
\bigcirc \quad alternate interior angles theorem
\bigcirc \quad corresponding angles theorem
\bigcirc \quad linear pair postulate
clear all

Explanation:

To determine which theorem justifies \( \angle 2 \cong \angle 5 \) when lines \( n \) and \( p \) are parallel, we analyze each option:

• Vertical Angles Theorem: Applies to angles opposite each other when two lines intersect, not related to parallel lines cut by a transversal for \( \angle 2 \) and \( \angle 5 \).
• Alternate Interior Angles Theorem: States that if two parallel lines are cut by a transversal, then alternate interior angles are congruent. \( \angle 2 \) and \( \angle 5 \) are alternate interior angles (they lie between the two parallel lines \( n \) and \( p \) and on opposite sides of the transversal), so this theorem applies.
• Corresponding Angles Theorem: Applies to angles in the same position relative to the parallel lines and transversal, which is not the case for \( \angle 2 \) and \( \angle 5 \).
• Linear Pair Postulate: Applies to adjacent angles forming a linear pair (supplementary), not relevant here.

Answer:

Alternate Interior Angles Theorem