A square can be defined as a parallelogram that has four congruent sides and right angles. However, this is the only definition. Determine which of the following definitions correctly define a square, explain why the definition is incorrect, and draw and label a counterexample shape that fits the definition but is not a parallelogram: 1. A parallelogram that has congruent sides and four right angles. 2. A parallelogram that has three congruent sides and four right angles. 3. A parallelogram that has two adjacent congruent sides and three right angles. 4. A parallelogram that has congruent diagonals.
The Correct Answer and Explanation is:
Let’s analyze each definition:
- A parallelogram that has congruent sides and four right angles.
- Correct Definition: This is indeed a correct definition of a square. If a parallelogram has four right angles and all sides congruent, it must be a square. This is because the properties of a parallelogram (opposite sides parallel and equal in length) combined with the additional conditions of right angles and congruent sides force the figure to be a square.
- A parallelogram that has three congruent sides and four right angles.
- Incorrect Definition: This is incorrect because in a parallelogram, opposite sides must be parallel and equal in length. A shape with three congruent sides cannot fulfill this condition. A parallelogram cannot have three congruent sides. If three sides are congruent, the fourth side would have to differ, breaking the definition of a parallelogram.
- Counterexample: A rectangle with three congruent sides could be drawn, where the opposite sides are parallel, and all angles are 90 degrees. However, this would not be a square since it would have unequal opposite sides.
- A parallelogram that has two adjacent congruent sides and three right angles.
- Incorrect Definition: This is also incorrect. A parallelogram cannot have three right angles because the fourth angle would also be a right angle due to the properties of parallelograms (the sum of adjacent angles equals 180 degrees). Therefore, this definition implies an impossible shape.
- Counterexample: A rectangle where only two sides are congruent could satisfy this, but it would not be a square. It would not fit the condition of having all four sides congruent.
- A parallelogram that has congruent diagonals.
- Incorrect Definition: This definition is incorrect because the property of congruent diagonals is not exclusive to squares. Rectangles also have congruent diagonals, but they are not squares unless all sides are congruent.
- Counterexample: A rectangle where opposite sides are congruent but not all sides. The diagonals of a rectangle are congruent, but it does not satisfy the square definition, as not all sides are congruent.
Summary of Definitions:
- Correct definition: Definition 1, where the parallelogram has congruent sides and four right angles.
- Incorrect definitions: Definitions 2, 3, and 4, as they either break basic parallelogram properties or fail to account for all the necessary conditions of a square.
