Question
Which biconditional statement is true?
A shape is a rectangle if and only if the shape has exactly four sides and four right angles.
A shape is a trapezoid if and only if the shape has a pair of parallel sides.
A shape is a triangle if and only if the shape has three sides and three acute angles.
A shape is a square if and only if the shape has exactly four congruent sides.
A shape is a rectangle if and only if the shape has exactly four sides and four right angles.
A shape is a trapezoid if and only if the shape has a pair of parallel sides.
A shape is a triangle if and only if the shape has three sides and three acute angles.
A shape is a square if and only if the shape has exactly four congruent sides.
Answers
GPT-4o mini
To evaluate the biconditional statements, we need to determine whether each statement accurately describes the conditions for the shapes mentioned.
1. **A shape is a rectangle if and only if the shape has exactly four sides and four right angles.**
- This statement is true. A rectangle always has four sides and four right angles. However, the converse is also true: if a shape has four sides and four right angles, then it is a rectangle.
2. **A shape is a trapezoid if and only if the shape has a pair of parallel sides.**
- This statement is also true. By definition, a trapezoid is a shape that has at least one pair of parallel sides. The converse holds: if a shape has one pair of parallel sides, it is considered a trapezoid.
3. **A shape is a triangle if and only if the shape has three sides and three acute angles.**
- This statement is false. A triangle must have three sides, but it does not have to have three acute angles. A triangle can have one right angle (right triangle) or one obtuse angle (obtuse triangle).
4. **A shape is a square if and only if the shape has exactly four congruent sides.**
- This statement is false. While a square does have four congruent sides, it also has four right angles. The converse is not true: a shape can have four congruent sides (a rhombus) but not be a square if it doesn't have right angles.
Based on this analysis, the true biconditional statements are:
- **A shape is a rectangle if and only if the shape has exactly four sides and four right angles.**
- **A shape is a trapezoid if and only if the shape has a pair of parallel sides.**
Both of these statements are true.
1. **A shape is a rectangle if and only if the shape has exactly four sides and four right angles.**
- This statement is true. A rectangle always has four sides and four right angles. However, the converse is also true: if a shape has four sides and four right angles, then it is a rectangle.
2. **A shape is a trapezoid if and only if the shape has a pair of parallel sides.**
- This statement is also true. By definition, a trapezoid is a shape that has at least one pair of parallel sides. The converse holds: if a shape has one pair of parallel sides, it is considered a trapezoid.
3. **A shape is a triangle if and only if the shape has three sides and three acute angles.**
- This statement is false. A triangle must have three sides, but it does not have to have three acute angles. A triangle can have one right angle (right triangle) or one obtuse angle (obtuse triangle).
4. **A shape is a square if and only if the shape has exactly four congruent sides.**
- This statement is false. While a square does have four congruent sides, it also has four right angles. The converse is not true: a shape can have four congruent sides (a rhombus) but not be a square if it doesn't have right angles.
Based on this analysis, the true biconditional statements are:
- **A shape is a rectangle if and only if the shape has exactly four sides and four right angles.**
- **A shape is a trapezoid if and only if the shape has a pair of parallel sides.**
Both of these statements are true.