To prove: A quadrilateral ABCD is a parallelogram.
Proof: It is given that ABCD is an equilateral, thus side AB is parallel to side DC so the alternate interior angles are congruent, thus ∠ABD and ∠BDC, are congruent.
Also, From ΔADB and ΔCDB, we have
∠ABD =∠BDC (Alternate angles)
Thus, by SAS rule of congruency, ΔADB is congruent to ΔCDB that is ΔADB≅ΔCDB.
By CPCTC, ∠DBC and ∠ADB are congruent and sides AD and BC are congruent. ∠DBC and ∠ADB form a pair of alternate interior angles.
Therefore, AD is congruent and parallel to BC.
Quadrilateral ABCD is a parallelogram because its opposite sides are equal and parallel.