Prove The Parallelogram Side Theorem

The Parallelogram Side Theorem states that in a parallelogram, the opposite sides are equal in length. To prove this theorem, we can use geometric properties and congruence.

Theorem Statement

In a parallelogram ABCDABCD:

• AB=CDAB = CD (opposite sides)
• AD=BCAD = BC (opposite sides)

Proof

Given: A parallelogram ABCDABCD
To Prove: AB=CDAB = CD and AD=BCAD = BC

Step 1: Draw the Parallelogram

Start by drawing parallelogram ABCDABCD with vertices AA, BB, CC, and DD.

Step 2: Use Triangle Congruence

1. Draw Diagonal: Draw diagonal ACAC.
2. Triangles Formed: This diagonal divides the parallelogram into two triangles: △ABC\triangle ABC and △CDA\triangle CDA.

Step 3: Show Triangles are Congruent

• Common Side: The side ACAC is common to both triangles △ABC\triangle ABC and △CDA\triangle CDA.
• Angle Correspondence: Since ABCDABCD is a parallelogram, the following angles are equal due to the properties of parallel lines:
  • ∠ABC=∠CDA\angle ABC = \angle CDA (Alternate interior angles)
  • ∠ACB=∠DAC\angle ACB = \angle DAC (Alternate interior angles)

Step 4: Apply the SAS (Side-Angle-Side) Congruence Criterion

• From the above points, we can conclude:
  • ACAC is common (side),
  • ∠ABC=∠CDA\angle ABC = \angle CDA (angle),
  • AB=CDAB = CD (sides of the triangles we want to prove).

Thus, by the SAS criterion, △ABC≅△CDA\triangle ABC \cong \triangle CDA.

Step 5: Conclusion from Triangle Congruence

Since the triangles △ABC\triangle ABC and △CDA\triangle CDA are congruent, the corresponding sides must also be equal:

• AB=CDAB = CD
• AD=BCAD = BC

Final Conclusion

Thus, we have proven that in a parallelogram ABCDABCD:

• AB=CDAB = CD and AD=BCAD = BC.

This completes the proof of the Parallelogram Side Theorem, confirming that opposite sides of a parallelogram are equal in length.