Question

given: ( w parallel x ) and ( y ) is a transversal. prove: ( angle 3 ) and ( angle 5 ) are supplementary. use the drop-down menus to complete the proof. given that ( w parallel x ) and ( y ) is a transversal, we know that ( angle 1 cong angle 5 ) by the drop-down. therefore, ( mangle 1 ) text. we also know that angle pair are a linear pair so they sum to 180. now we can substitute ( + mangle 5 = 180 ). therefore, by the definition of supplementary angles, ( angle 3 ) and ( angle 5 ) are supplementary. drop-down options: corresponding angles theorem, alternate interior angles theorem, vertical angles theorem, alternate exterior angles theorem. diagram: lines ( w, x ) (parallel), transversal ( y ); angles ( 1,2,3,4 ) (at ( w ) and ( y )), ( 5,6,7,8 ) (at ( x ) and ( y )).

Explanation:

Step1: Identify the theorem for ∠1 ≅ ∠5

Since \( w \parallel x \) and \( y \) is a transversal, \( \angle 1 \) and \( \angle 5 \) are corresponding angles. The corresponding angles theorem states that if two parallel lines are cut by a transversal, then each pair of corresponding angles is congruent. So \( \angle 1 \cong \angle 5 \) by the corresponding angles theorem.

Step2: Recall the linear pair property

\( \angle 1 \) and \( \angle 3 \) form a linear pair (they are adjacent and form a straight line), so \( m\angle 1 + m\angle 3 = 180^\circ \) (by the definition of a linear pair, which are supplementary).

Step3: Substitute \( m\angle 1 \) with \( m\angle 5 \)

Since \( \angle 1 \cong \angle 5 \), their measures are equal, so \( m\angle 1 = m\angle 5 \). Substituting \( m\angle 1 \) with \( m\angle 5 \) in \( m\angle 1 + m\angle 3 = 180^\circ \), we get \( m\angle 3 + m\angle 5 = 180^\circ \). By the definition of supplementary angles, \( \angle 3 \) and \( \angle 5 \) are supplementary.

Answer:

The first drop - down should be filled with "corresponding angles theorem".