Question
Is the following biconditional true? A quadrilateral is an isosceles trapezoid if and only if it has a pair of opposite congruent sides. yes
Ask by Harmon Lang. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The biconditional statement is not true because a quadrilateral with a pair of opposite congruent sides is not necessarily an isosceles trapezoid.
Solution
To determine if the biconditional statement "A quadrilateral is an isosceles trapezoid if and only if it has a pair of opposite congruent sides" is true, we need to analyze both directions of the biconditional:
1. **If a quadrilateral is an isosceles trapezoid, then it has a pair of opposite congruent sides.**
- By definition, an isosceles trapezoid is a trapezoid (a quadrilateral with at least one pair of parallel sides) where the non-parallel sides (the legs) are congruent. Therefore, it has at least one pair of opposite congruent sides (the legs).
2. **If a quadrilateral has a pair of opposite congruent sides, then it is an isosceles trapezoid.**
- This statement is not necessarily true. A quadrilateral can have a pair of opposite congruent sides without being an isosceles trapezoid. For example, a rectangle has two pairs of opposite congruent sides but is not classified as an isosceles trapezoid.
Since the second part of the biconditional is false, the entire biconditional statement is false.
Thus, the answer is **no**, the biconditional is not true.
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Mind Expander
The statement is indeed true! An isosceles trapezoid is defined as a trapezoid with at least one pair of opposite sides that are both parallel and congruent. This uniqueness means that if you have a quadrilateral with one pair of opposite congruent sides, it can be classified as an isosceles trapezoid. However, it's worth noting that while all isosceles trapezoids possess a pair of congruent sides, the converse is not universally true—some quadrilaterals might have a pair of congruent sides without being isosceles trapezoids, like certain parallelograms. So, it's essential to consider the full shape to ensure it adheres to the definition of an isosceles trapezoid!
