TIME REMA 47:0 BC || AD and CD || BA because the Therefore, ABCD is a parallelogram because both pairs of opposite sides are parallel. Olengths of consecutive sides are not equal slopes of opposite sides are equal Olengths of opposite sides are equal slopes of consecutive sides are not equal
The Correct Answer and Explanation is:
The correct answer is:
Slopes of opposite sides are equal
Therefore, ABCD is a parallelogram because both pairs of opposite sides are parallel.
Explanation
A parallelogram is a quadrilateral where both pairs of opposite sides are parallel. The most direct way to prove that a figure is a parallelogram is by using the slopes of its sides. In coordinate geometry, if two line segments have the same slope, they are parallel.
In the context of this question:
- You are told that REMA 47:0, which is likely a reference to a coordinate system or problem number.
- You are given that BC || AD and CD || BA.
- These notations imply that side BC is parallel to side AD, and side CD is parallel to side BA.
In coordinate geometry, this means:
- The slope of BC = slope of AD
- The slope of CD = slope of BA
Since both pairs of opposite sides have equal slopes, they are parallel, satisfying the definition of a parallelogram.
Let’s break down why other options are incorrect:
- “Lengths of consecutive sides are not equal”: This may be true, but it’s irrelevant. A parallelogram does not require consecutive sides (like AB and BC) to be equal—only opposite sides.
- “Slopes of consecutive sides are not equal”: This is also not a requirement. In fact, we expect slopes of consecutive sides (like AB and BC) to differ unless it’s a rectangle or square.
- “Lengths of opposite sides are equal”: While this is a property of parallelograms, it is not a valid proof on its own unless both pairs are shown to be equal. The slope (parallelism) is a stronger and more direct geometric proof.
Conclusion:
Since both pairs of opposite sides have equal slopes, they are parallel. Therefore, ABCD is a parallelogram because both pairs of opposite sides are parallel.
