Final answer:
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:
- Since AC and BD are perpendicular and congruent, triangles ABC and ACD are right triangles and have a side length in common, AC = BD.
- 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.
- Therefore, all sides of the quadrilateral ABCD are congruent, making it a square.