Proof: Diagonals of a parallelogram bisect each other | Quadrilaterals | Geometry | Khan Academy

Introduction

This tutorial provides a step-by-step guide to proving that the diagonals of a parallelogram bisect each other. Understanding this property is essential in geometry, particularly when working with quadrilaterals. This proof will help solidify your knowledge of parallelograms and their characteristics.

Step 1: Understand the Definition of a Parallelogram

Before proceeding with the proof, familiarize yourself with the properties of a parallelogram:

• A parallelogram is a four-sided figure (quadrilateral) where opposite sides are parallel.
• The opposite sides are equal in length.
• The opposite angles are equal.

Step 2: Set Up the Diagram

To visually represent the proof:

1. Draw a parallelogram labeled as ABCD.
2. Mark the diagonals AC and BD.
3. Label the intersection point of the diagonals as point E.

Step 3: Apply Triangle Properties

Use the triangle properties to establish relationships:

• Triangles ABE and CDE are formed by the diagonals.
• Since AB is parallel to CD and AE is a transversal, angle ABE is congruent to angle CDE (Alternate Interior Angles Theorem).

Step 4: Prove Triangle Congruence

To show that triangles ABE and CDE are congruent:

1. Sides:

   • AE is a common side.
   • AB = CD (opposite sides of a parallelogram are equal).
   • BE = DE (as we will prove that diagonals bisect each other).

2. Angles:

   • Angle ABE = Angle CDE (from Step 3).

Using the Side-Angle-Side (SAS) postulate, we can conclude that triangle ABE is congruent to triangle CDE.

Step 5: Conclude the Proof

From the congruence of triangles ABE and CDE:

• The corresponding parts of congruent triangles are equal, which means:
  • BE = DE.
• Since point E is the midpoint of both diagonals, this confirms that diagonals AC and BD bisect each other.

Conclusion

In this tutorial, we established that the diagonals of a parallelogram bisect each other through a systematic proof involving triangle properties and congruence. Understanding this theorem is a stepping stone for further studies in geometry, particularly in the properties of quadrilaterals. For more advanced topics, consider exploring the properties of opposite angles in parallelograms or exploring specific types of quadrilaterals.