Question

to prove quadrilateral wxyz is a parallelogram, travis begins by proving △wzy ≅ △yxw by using the sas congruency theorem. which reasons can travis use to prove the two triangles are congruent? check all that apply. ∠zwy ≅ ∠xyw by the alternate interior ∠s theorem. (overline{wy}) ≅ (overline{wy}) by the reflexive property. ∠zwy ≅ ∠xwy by the corresponding ∠s theorem. (overline{wx}) ≅ (overline{zy}) by definition of a parallelogram. (overline{wz}) ≅ (overline{xy}) by the given.

Explanation:

• For \( \angle ZWY\cong\angle XYW\):
• If \( WZ\parallel XY\) (property of a parallelogram which we are trying to prove, but in the context of using alternate - interior angles for congruent triangles), when a transversal \( WY\) intersects two parallel lines \( WZ\) and \( XY\), the alternate - interior angles \( \angle ZWY\) and \( \angle XYW\) are congruent by the alternate - interior angles theorem.
• For \( \overline{WY}\cong\overline{WY}\):
• By the reflexive property of congruence, any segment is congruent to itself. So, \( \overline{WY}\) is congruent to \( \overline{WY}\).
• For \( \overline{WX}\cong\overline{ZY}\):
• In a parallelogram, opposite sides are congruent. Since we are proving \( WXYZ\) is a parallelogram, but if we assume the given figure has markings (from the side - marking in the diagram) that \( WX\) and \( ZY\) are congruent (by the definition of a parallelogram, which is what we want to prove, but in the context of triangle congruence for parallelogram proof, if we consider the side - side - side or side - angle - side relations).
• For \( \overline{WZ}\cong\overline{XY}\):
• If it is given (from the side - marking in the diagram) that \( WZ\) and \( XY\) are congruent.

• For \( \angle ZWY\cong\angle XWY\):
• There is no reason (such as parallel lines with a proper transversal) to claim \( \angle ZWY\cong\angle XWY\) by the corresponding angles theorem. Corresponding angles are formed when a transversal intersects two parallel lines, and there is no indication that the lines forming these angles (in the way required for corresponding angles) are parallel.

Answer:

\(\angle ZWY\cong\angle XYW\) by the alternate interior \(\angle s\) theorem, \(\overline{WY}\cong\overline{WY}\) by the reflexive property, \(\overline{WX}\cong\overline{ZY}\) by definition of a parallelogram, \(\overline{WZ}\cong\overline{XY}\) by the given.