Question

given: e || f and g is a transversal
prove: ∠1 ≅ ∠8

given that e || f and g is a transversal, we know that
∠4 ≅ ∠5 by the alternate interior angles theorem. we
also know that ∠1 ≅ ∠4 and ∠5 ≅ ∠8 by the ______.
therefore, ∠1 ≅ ∠8 by the substitution property.

○ corresponding angles theorem
○ alternate interior angles theorem
○ vertical angles theorem
○ alternate exterior angles theorem

Explanation:

To determine the theorem for \( \angle 1 \cong \angle 4 \) and \( \angle 5 \cong \angle 8 \), we analyze each option:

• Corresponding angles theorem: Applies to angles in the same position relative to parallel lines and a transversal, not vertical angles. Eliminate.
• Alternate interior angles theorem: Applies to non - adjacent interior angles, not vertical angles. Eliminate.
• Vertical angles theorem: Vertical angles (opposite angles formed by intersecting lines) are congruent. \( \angle 1 \) and \( \angle 4 \) are vertical angles, \( \angle 5 \) and \( \angle 8 \) are vertical angles. This fits.
• Alternate exterior angles theorem: Applies to non - adjacent exterior angles, not vertical angles. Eliminate.

Answer:

C. vertical angles theorem