Which of the following statements is true? All trapezoids are parallelograms. All parallelograms are trapezoids. All rectangles are squares. All rhombuses are rectangles.
The correct answer and explanation is:
The correct answer is: None of the statements are true.
Here’s an explanation for each statement:
- All trapezoids are parallelograms: This is false. A trapezoid is a quadrilateral with only one pair of parallel sides. A parallelogram, on the other hand, is a quadrilateral where opposite sides are parallel and equal in length. Not all trapezoids meet the requirements of parallelograms.
- All parallelograms are trapezoids: This is also false. As mentioned, a parallelogram has both pairs of opposite sides parallel, whereas a trapezoid has only one pair of parallel sides. Therefore, not all parallelograms are trapezoids.
- All rectangles are squares: This is false. A rectangle is a quadrilateral with four right angles, and opposite sides equal in length. A square is a special type of rectangle where all four sides are equal in length. So, while all squares are rectangles, not all rectangles are squares.
- All rhombuses are rectangles: This is false. A rhombus is a quadrilateral where all sides are of equal length, but its angles are not necessarily 90 degrees. A rectangle has four right angles, so while all rectangles are parallelograms, not all rhombuses are rectangles because their angles may not be right angles.
In summary, each of these statements contains an overgeneralization that does not hold true when considering the properties and definitions of each geometric shape.