Question

which statement explains how you could use coordinate geometry to prove that quadrilateral abcd is a parallelogram? prove that all sides are congruent and have slopes that are opposite reciprocals. prove that segments ad and ab are congruent and parallel. prove that opposite sides are congruent and have equal slopes. prove that segments bc and cd are congruent and parallel.

Explanation:

Step1: Recall the definition of a parallelogram

A parallelogram is a quadrilateral with both pairs of opposite sides parallel.

Step2: Analyze the properties of sides in coordinate geometry

In coordinate geometry, if two line segments have the same slope, they are parallel. Also, if the lengths of opposite sides (calculated using the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)) are equal (congruent) and their slopes are equal (for parallelism), then the quadrilateral is a parallelogram.

Step3: Evaluate each option

• Option 1 (Prove that segments BC and CD are congruent and parallel): This only focuses on two adjacent sides, not the opposite sides which is the key property of a parallelogram.
• Option 2 (Prove that opposite sides are congruent and have equal slopes): This is correct. If opposite sides are congruent (equal in length) and have equal slopes (parallel), by the definition of a parallelogram (a quadrilateral with both pairs of opposite sides parallel), the quadrilateral is a parallelogram.
• Option 3 (Prove that segments AD and AB are congruent and parallel): This focuses on two adjacent sides, not the opposite sides.
• Option 4 (Prove that all sides are congruent and have slopes that are opposite reciprocals): If slopes are opposite reciprocals, the sides are perpendicular (which is a property of a rhombus - square in some cases, not the general parallelogram property. Also, a parallelogram only requires opposite sides to be congruent, not all sides.

Answer:

Prove that opposite sides are congruent and have equal slopes.