Final answer:
To prove △VWY≡△VXY given the conditions, we utilize the midpoint definition and the Hypotenuse-Leg (HL) congruence theorem for right triangles, ultimately showing congruence via side-angle-side (SAS).
Step-by-step explanation:
To prove that △VWY≡△VXY when Y is the midpoint of WX and WX ⟂ VY, we follow a series of logical steps involving the properties of triangles and the definition of a midpoint. Since Y is the midpoint of WX, we know by definition that WY = XY. Given the perpendicularity, ∠VYW and ∠VXY are both right angles, following from the definition of perpendicular lines. Therefore, we have two pairs of sides that are congruent (WY = XY, and VY is common to both triangles) and a pair of congruent angles (∠VYW = ∠VXY).
By the Hypotenuse-Leg (HL) congruence theorem for right triangles, if the hypotenuse and one leg of one right triangle are congruent to the hypotenuse and one leg of another right triangle, then the triangles are congruent. In our case, we have VY as the hypotenuse and WY = XY as a leg for both △VWY and △VXY. This allows us to conclude that △VWY≡△VXY by the HL theorem.
In summary, we have shown that two sides and the angle between them (commonly referred to as side-angle-side, or SAS) are congruent in each triangle. Since one of the congruent angles is a right angle, the HL theorem specifically applies, completing the proof of △VWY≡△VXY.