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Which statement below is false? () all parallelograms are also trapezoids (2) all squares are also rectangles (3) all rhombuses are also squares (4) all rectangles are also parallelograms

Ask by Kirk Wang. in the United States
Mar 10,2025

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Answer

The false statement is: all rhombuses are also squares.

Solution

\( \textbf{Step 1: Analyze Statement (1)} \) A trapezoid is commonly defined as a quadrilateral with at least one pair of parallel sides. Since every parallelogram has two pairs of parallel sides, every parallelogram qualifies as a trapezoid under this definition. \( \textbf{Step 2: Analyze Statement (2)} \) A square has all sides equal and all angles equal to \(90^\circ\). A rectangle only requires that all angles are \(90^\circ\). Therefore, every square meets the criteria for being a rectangle. \( \textbf{Step 3: Analyze Statement (3)} \) A rhombus has all sides equal but does not necessarily have all angles equal to \(90^\circ\). For a rhombus to be a square, it must also have all angles equal to \(90^\circ\). Since this is not true for every rhombus, the statement that all rhombuses are squares is false. \( \textbf{Step 4: Analyze Statement (4)} \) A rectangle has two pairs of parallel sides, which means it fits the definition of a parallelogram. Therefore, every rectangle is a parallelogram. \( \textbf{Conclusion} \) The false statement is: \( \boxed{\text{(3) all rhombuses are also squares}} \)

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The Deep Dive

Did you know that the relationship between different quadrilaterals has a fascinating historical aspect? The study of these shapes dates back to ancient civilizations, where mathematicians like Euclid laid the groundwork for geometry. They explored various properties, such as angles and sides, that led to the classifications we use today in modern geometry. In the real world, understanding the properties of shapes can be applied in architecture and design. Architects utilize the unique characteristics of parallelograms, trapezoids, and rectangles to create visually appealing structures. The angles and side relations help in ensuring stability, aesthetic appeal, and functionality in buildings and furniture designs.

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