Question

Prove: If the diagonals of a quadrilateral are perpendicular and congruent, then the quadrilateral is a square.

Answer 1

A quadrilateral with congruent and perpendicular diagonals has the properties of a rectangle and a rhombus, which means it is a square, as it has equal sides and all right angles.

Step-by-step explanation:

To prove that if the diagonals of a quadrilateral are perpendicular and congruent, then the quadrilateral is a square, we can use the properties of a square, such as its diagonals being congruent and perpendicular to each other.

Suppose we have a quadrilateral ABCD with perpendicular diagonals AC and BD, and AC = BD.

To prove that the quadrilateral is a square, we need to show that all sides are congruent and all angles are right angles.

Proof:

1. Since AC and BD are perpendicular and congruent, triangles ABC and ACD are right triangles and have a side length in common, AC = BD.
2. Since triangles ABC and ACD share a common side and have two pairs of congruent angles (since their respective sides are parallel), they are congruent by Side-Angle-Side (SAS) congruence.
3. Therefore, all sides of the quadrilateral ABCD are congruent, making it a square.