We saw that one of the properties of a rectangle is that its opposite sides are parallel. Can this be chosen as a definition of a rectangle? In other words, is every quadrilateral that has opposite sides parallel and equal, a rectangle?
No, a quadrilateral with opposite sides parallel and equal is not necessarily a rectangle.
Explanation: A quadrilateral with opposite sides parallel and equal is called a parallelogram.
A parallelogram has opposite sides parallel and equal, but its angles need not be 90°.
A rectangle is a special type of parallelogram where all angles are 90°.
Conclusion: ∴ Having opposite sides parallel and equal is not sufficient to define a rectangle—the condition that all angles must be 90° is also required.
Explanation
The chapter introduces parallelograms in Section 4.3 as quadrilaterals with parallel opposite sides. It states that a parallelogram has opposite sides equal (Property 1) and parallel (Property 2), but the angles need not be 90°. A rectangle is defined as a special kind of parallelogram with all angles equal to 90°. Thus, the definition of a rectangle requires both the parallelogram property AND the right angle property.
Solution Steps
Identify that a quadrilateral with opposite sides parallel and equal is called a parallelogram.
Note that a parallelogram's angles need not be 90°.
Recognize that a rectangle is a special type of parallelogram with all angles equal to 90°.
Conclude that having opposite sides parallel and equal alone is not sufficient to define a rectangle.