Question

given: w ∥ x and y is a transversal. prove: ∠3 and ∠5 are supplementary. use the drop - down menus to complete the proof. given that w ∥ x and y is a transversal, we know that ∠1 ≅ ∠5 by the drop - down menu. therefore, m∠1 content. we also know that ∠1 and ∠3 form a linear pair so they are supplementary, so m∠1 + m∠3 = 180. now we can substitute m∠1 with m∠5 to get m∠3 + m∠5 = 180. therefore, by the definition of supplementary angles, ∠3 and ∠5 are supplementary. the options in the drop - down menu are: corresponding angles theorem, alternate interior angles theorem, vertical angles theorem, alternate exterior angles theorem.

Explanation:

Step1: Identify Angle Relationship

∠1 and ∠5 are corresponding angles (same position relative to parallel lines \( w \parallel x \) and transversal \( y \)). The corresponding angles theorem states that if two parallel lines are cut by a transversal, corresponding angles are congruent. So \( \angle 1 \cong \angle 5 \) by the corresponding angles theorem.

Step2: Use Linear Pair Property

∠1 and ∠3 form a linear pair (adjacent angles on a straight line), so their measures sum to \( 180^\circ \), i.e., \( m\angle 1 + m\angle 3 = 180^\circ \).

Step3: Substitute Congruent Angles

Since \( \angle 1 \cong \angle 5 \), \( m\angle 1 = m\angle 5 \). Substituting \( m\angle 1 \) with \( m\angle 5 \) in \( m\angle 1 + m\angle 3 = 180^\circ \) gives \( m\angle 5 + m\angle 3 = 180^\circ \), proving \( \angle 3 \) and \( \angle 5 \) are supplementary.

Answer:

corresponding angles theorem