Question

in the given figure x = 145°. what theorem can be used to prove the lines p and q are parallel? a) alternate exterior angles converse theorem b) same side interior angles converse theorem c) alternate interior angles converse theorem d) corresponding angles converse theorem

Explanation:

To determine the theorem, we analyze the angles formed by the transversal with lines \( p \) and \( q \). The angle \( x = 145^\circ \) and the \( 145^\circ \) angle on line \( p \) are corresponding angles (they occupy the same relative position at each intersection). The Corresponding Angles Converse Theorem states that if corresponding angles are congruent, then the lines are parallel. So, we check the options:

• Option A: Alternate exterior angles are on opposite sides and outside the lines, not matching here.
• Option B: Same - side interior angles are on the same side and inside, not applicable.
• Option C: Alternate interior angles are on opposite sides and inside, not the case here.
• Option D: Corresponding angles converse theorem applies as the angles are corresponding and congruent, proving \( p\parallel q \).

Answer:

D) Corresponding angles converse theorem