Given that WXYZ is a rhombus with angle X = 90 degrees, what statement proves that WXYZ is a square?

A. All sides of a rhombus are congruent.
B. A rhombus has diagonals that bisect each other at right angles.
C. A rhombus is a parallelogram with all sides congruent.
D. A rhombus is a rectangle with diagonals congruent.

Question

28-08-2024
Mathematics

High School

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Given that WXYZ is a rhombus with angle X = 90 degrees, what statement proves that WXYZ is a square?

A. All sides of a rhombus are congruent.
B. A rhombus has diagonals that bisect each other at right angles.
C. A rhombus is a parallelogram with all sides congruent.
D. A rhombus is a rectangle with diagonals congruent.

Answer :

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Final answer:

Statement (d) 'A rhombus is a rectangle with diagonals congruent' proves that a rhombus with angle X equals 90 degrees is a square, since having right angles and congruent diagonals satisfies the definition of a square. So the final answer is option d.

Explanation:

Given that WXYZ is a rhombus with angle X equal to 90 degrees, the statement that proves that WXYZ is a square is option (d): 'A rhombus is a rectangle with diagonals congruent.' All sides of a rhombus are congruent. However, the defining characteristic that makes a rhombus a square is having all four angles as right angles.

Since we are given that angle X is 90 degrees, and we know that a rhombus has diagonals that bisect each other at right angles (making it a parallelogram), if we can also establish that the rhombus has diagonals that are congruent, then it must be a square. Hence, statement (d) that a rhombus is a rectangle (which by definition has all right angles).

With diagonals congruent confirms that our rhombus is actually a square. Statement (d) 'A rhombus is a rectangle with diagonals congruent' proves that a rhombus with angle X equals 90 degrees is a square, since having right angles and congruent diagonals satisfies the definition of a square.