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Answer :
Certainly! Let's analyze each statement given in the question and determine which are true based on the properties of quadrilaterals with congruent diagonals.
1. If you also knew that WY and XZ were perpendicular, then you could prove WXYZ is a square.
- This would imply that not only the diagonals are congruent but also perpendicular. If the diagonals of a quadrilateral are both congruent and perpendicular, and all sides are equal, then WXYZ could be a square. For a square, the diagonals must be equal and perpendicular, forming right angles at their intersection, which is a property exclusive to squares and rhombuses.
2. If you also knew that WZ and XY were congruent, then you could prove WXYZ is a rectangle.
- In a rectangle, opposite sides are equal in length. So if WZ and XY are congruent, combined with the knowledge that the diagonals are congruent, you could potentially have a rectangle. Congruent diagonals are a unique feature of rectangles and squares.
3. If you also knew that WY and XZ bisected each other, then you could prove WXYZ is a rectangle.
- When diagonals bisect each other, it suggests a parallelogram. In this case, if the diagonals are also congruent, then WXYZ could indeed be a rectangle. In a rectangle, the diagonals not only bisect each other but are also congruent.
4. There is enough information to prove that WXYZ is a parallelogram.
- Merely having congruent diagonals does not provide enough information to prove that WXYZ is a parallelogram. For a quadrilateral to be a parallelogram, the diagonals need to bisect each other, among other properties. Just congruence of the diagonals is insufficient without additional information about side lengths or angles.
Based on these analyses, the statements that are true are:
- If WY and XZ were perpendicular, WXYZ could be a square if all sides are equal.
- If WZ and XY were congruent, WXYZ could be a rectangle.
- If WY and XZ bisected each other, WXYZ could be a rectangle.
Therefore, the true statements are: First, Second, and Third statements.
1. If you also knew that WY and XZ were perpendicular, then you could prove WXYZ is a square.
- This would imply that not only the diagonals are congruent but also perpendicular. If the diagonals of a quadrilateral are both congruent and perpendicular, and all sides are equal, then WXYZ could be a square. For a square, the diagonals must be equal and perpendicular, forming right angles at their intersection, which is a property exclusive to squares and rhombuses.
2. If you also knew that WZ and XY were congruent, then you could prove WXYZ is a rectangle.
- In a rectangle, opposite sides are equal in length. So if WZ and XY are congruent, combined with the knowledge that the diagonals are congruent, you could potentially have a rectangle. Congruent diagonals are a unique feature of rectangles and squares.
3. If you also knew that WY and XZ bisected each other, then you could prove WXYZ is a rectangle.
- When diagonals bisect each other, it suggests a parallelogram. In this case, if the diagonals are also congruent, then WXYZ could indeed be a rectangle. In a rectangle, the diagonals not only bisect each other but are also congruent.
4. There is enough information to prove that WXYZ is a parallelogram.
- Merely having congruent diagonals does not provide enough information to prove that WXYZ is a parallelogram. For a quadrilateral to be a parallelogram, the diagonals need to bisect each other, among other properties. Just congruence of the diagonals is insufficient without additional information about side lengths or angles.
Based on these analyses, the statements that are true are:
- If WY and XZ were perpendicular, WXYZ could be a square if all sides are equal.
- If WZ and XY were congruent, WXYZ could be a rectangle.
- If WY and XZ bisected each other, WXYZ could be a rectangle.
Therefore, the true statements are: First, Second, and Third statements.
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