Difficulty: Hard

Average Score: 39%

If a quadrilateral is a rectangle, which of the following statements must be true? Select ALL that apply.

Rationale

The quadrilateral has four congruent angles, two pairs of parallel sides, and the two diagonals have equal length.

A rectangle is defined as a quadrilateral with four right angles, two pairs of opposite sides that are parallel, and equal-length diagonals. These properties are inherent to all rectangles and distinguish them from other quadrilaterals.

A (The quadrilateral has four congruent sides.) YOUR ANSWER

This statement is false because a rectangle does not require all four sides to be congruent; only opposite sides need to be equal in length. A square is a specific type of rectangle that has all sides congruent, but not all rectangles share this property.

B (The quadrilateral has four congruent angles.) CORRECT

This statement is true as all angles in a rectangle are right angles, measuring 90 degrees. Thus, the angles are congruent, fulfilling the definition of a rectangle.

C (The quadrilateral has two pairs of parallel sides.) CORRECT

This statement holds true since a rectangle is defined as having two pairs of opposite sides that are parallel. This property is essential for the classification of a quadrilateral as a rectangle.

D (The two diagonals of the quadrilateral have equal length.) CORRECT

This statement is also true. In a rectangle, the diagonals are equal in length due to the properties of congruent triangles formed by the diagonals, confirming that this is a characteristic of rectangles.

E (The two diagonals of the quadrilateral are perpendicular to each other.) YOUR ANSWER

This statement is false because while diagonals in a rectangle are equal, they are not necessarily perpendicular. This property is only true for special quadrilaterals like rhombuses or squares.

Conclusion

A rectangle must have four congruent angles, two pairs of parallel sides, and equal-length diagonals, which are essential properties of this shape. The incorrect options highlight common misconceptions, emphasizing that a rectangle's defining characteristics do not include congruent side lengths or perpendicular diagonals. Understanding these distinctions is vital in geometry for correctly identifying and classifying quadrilaterals.

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