Answer :
To determine whether Mary's statement is true or false, let's analyze the properties of a parallelogram and the implications of congruent diagonals.
First, we know that a parallelogram is a quadrilateral with opposite sides that are parallel and equal in length. Key properties of a parallelogram include:
1. Opposite sides are equal.
2. Opposite angles are equal.
3. Consecutive angles are supplementary.
4. Diagonals bisect each other.
Now, let's consider Mary's statement: "If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle."
A rectangle is a special type of parallelogram where:
1. All angles are right angles (each 90 degrees).
2. The diagonals are equal in length or congruent.
Since a rectangle is a specific type of parallelogram, if we can show that a parallelogram with congruent diagonals necessarily has all right angles, we can directly conclude that it is indeed a rectangle.
For a parallelogram, consider the two diagonals. If these diagonals are congruent (equal in length):
- Each diagonal will divide the parallelogram into two congruent triangles (by the Side-Side-Side postulate).
Since the triangles are congruent and the corresponding parts of congruent triangles are equal (CPCTC), the angles within the parallelogram will be equal.
For these internal angles to divide the parallelogram into accurate geometric shapes, each angle must be a right angle (90 degrees). This is because the congruency of the diagonals enforces symmetry, making each interior angle equal, and the only feasible solution for equal angles adding up to 360 degrees (the sum of interior angles in a parallelogram) is when each angle is 90 degrees.
Thus, a parallelogram with congruent diagonals must be a rectangle.
Therefore, Mary's statement is true, so the correct choice is:
A. True
First, we know that a parallelogram is a quadrilateral with opposite sides that are parallel and equal in length. Key properties of a parallelogram include:
1. Opposite sides are equal.
2. Opposite angles are equal.
3. Consecutive angles are supplementary.
4. Diagonals bisect each other.
Now, let's consider Mary's statement: "If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle."
A rectangle is a special type of parallelogram where:
1. All angles are right angles (each 90 degrees).
2. The diagonals are equal in length or congruent.
Since a rectangle is a specific type of parallelogram, if we can show that a parallelogram with congruent diagonals necessarily has all right angles, we can directly conclude that it is indeed a rectangle.
For a parallelogram, consider the two diagonals. If these diagonals are congruent (equal in length):
- Each diagonal will divide the parallelogram into two congruent triangles (by the Side-Side-Side postulate).
Since the triangles are congruent and the corresponding parts of congruent triangles are equal (CPCTC), the angles within the parallelogram will be equal.
For these internal angles to divide the parallelogram into accurate geometric shapes, each angle must be a right angle (90 degrees). This is because the congruency of the diagonals enforces symmetry, making each interior angle equal, and the only feasible solution for equal angles adding up to 360 degrees (the sum of interior angles in a parallelogram) is when each angle is 90 degrees.
Thus, a parallelogram with congruent diagonals must be a rectangle.
Therefore, Mary's statement is true, so the correct choice is:
A. True
