Question

The diagonals of quadrilateral WXYZ intersect at R. If R is the midpoint of WY and XZ, which additional statement shows that WXYZ is a SQUARE

1. WY ≌ XZ
2.∠ WZY=90°
3. WY⊥ xZ
4. a and c
a. a
b. b
c. c
d. d

Answer 1

To confirm that WXYZ is a square given that R is the midpoint of both diagonals, the additional information needed is that the diagonals are congruent and perpendicular. The correct choice is d: Options 1 and 3.

Step-by-step explanation:

To determine if quadrilateral WXYZ is a square, we can analyze the given options. For a quadrilateral to be a square, it must have equal sides, equal diagonals, and all angles must be 90 degrees.

• Option 1: WY ≅ XZ. If both diagonals are congruent, it supports the idea of a square, but it is not sufficient alone since rhombuses also have this property.
• Option 2: ∠ WZY=90°. A single 90-degree angle does not ensure all angles are 90 degrees and that sides are equal.
• Option 3: WY⊥XZ. If the diagonals are perpendicular, this suggests a square or a rhombus, but, like option 1, alone it is not enough.
• Options 1 and 3: WY ≅ XZ and WY⊥XZ. When the diagonals are both congruent and perpendicular, it indicates a square because a rhombus would not necessarily have congruent diagonals.

Therefore, given that R is the midpoint of both diagonals (WY and XZ), the additional information needed to conclude that WXYZ is a square is that both diagonals are congruent and perpendicular to each other. So, the correct option is d: Choices 1 and 3.