Quadrilateral ABCD is a Parallelogram

The theorem that the quadrilateral formed by joining the consecutive midpoints of another quadrilateral is a parallelogram has always fascinated geometry enthusiasts. This surprising fact can be revealed through careful study and application of various geometric concepts.

To prove that quadrilateral ABCD is a parallelogram, we will use the definition of a parallelogram as a quadrilateral with two pairs of opposite sides that are parallel to each other. We will show that this condition holds true for ABCD by demonstrating that both pairs of opposite sides are parallel.

First, we know that AD || BC, which means that the corresponding angles are congruent. This can be proven using the converse Alternate Interior Angles Theorem. By creating two triangles with equal sides and a common diagonal BD, we can show that ∠ADB ≅ ∠CBD as alternating interior angles formed by a transversal line.

Next, we will focus on showing that AB || DC. To do this, we will create two congruent triangles using the Side-Angle-Side postulate. By applying CPCTC (corresponding parts of congruent triangles are congruent), we can conclude that ∠BDC ≅ ∠DBA.

Now, we can apply the converse Alternate Interior Angles Theorem to show that AB || DC. This proves that one pair of opposite sides is parallel. Since we already know that AD || BC, we have demonstrated that both pairs of opposite sides are parallel, making ABCD a parallelogram.

This theorem highlights the importance of careful study in Euclidean Geometry, as it reveals surprising facts and properties that can be used to solve problems and prove theorems. By applying geometric concepts and theorems, we can gain a deeper understanding of the relationships between different shapes and angles.

Quadrilateral abcd is a parallelogram