Question
10. Which polygon does not always have congruent diagonals? \( \begin{array}{ll}\text { (1) square } & \text { (3) rhombus } \\ \text { (2) rectangle } & \text { (4) isosceles trapezoid }\end{array} \)
Ask by Valdez Mullins. in the United States
Mar 20,2025
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Tutor-Verified Answer
Answer
A rhombus does not always have congruent diagonals.
Solution
1. In a **square**, all angles are \(90^\circ\) and the diagonals are congruent.
2. In a **rectangle**, while the diagonals are not perpendicular, they are congruent.
3. In a **rhombus**, the diagonals are perpendicular bisectors; however, they are not necessarily congruent unless it is a square — that is, a special case.
4. In an **isosceles trapezoid**, the diagonals are congruent.
Thus, the polygon that does not always have congruent diagonals is the **rhombus**.
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An isosceles trapezoid typically has one pair of congruent sides and one pair of congruent angles, which helps in establishing a certain symmetry. However, the diagonals of an isosceles trapezoid do not always have the same length unless they are specifically set to do so, making it the only option here where diagonals aren’t always congruent. In contrast, both rectangles and squares have diagonals that are always equal, as do rhombuses, due to their unique properties. This makes those shapes reliable when it comes to diagonal congruence—a fun fact if you’re sketching polygons or calculating their properties!
