Lauren made the plan shown for proving that quadrilateral ABCD with

AB ≅ CD and BC ≅ DA is a parallelogram, by showing that the opposite angles are congruent. Plan: Draw in diagonals AC and BD. The given information and the shared side AC along with the Reflexive Property can be used to prove by the SSS Congruence Postulate. Using CPCTC, . The same can be done for using the given information and the shared side BD¯¯¯¯¯¯¯¯. This will lead to . Therefore, ABCD is a parallelogram because opposite angles are congruent.

It is given that the quadrilateral ABCD has AB ≅ CD and BC ≅ DA is a parallelogram, then in order to prove opposite angles of the parallelogram are equal, we take ΔABC and ΔADC,

AC=AC(Common)

AB=CD(given)

BC=AD(given)

Thus, by SSS rule, ΔABC ≅ ΔADC

By CPCT, ∠B=∠C

Also, from ΔABD and ΔBCD, we have

AB=CD(given)

BC=AD(given)

BD=BD(common)

Thus, by SSS rule, ΔABD ≅ ΔBCD

By CPCT, ∠A=∠C

Since, opposite angles are equal,therefore ABCD is a parallelogram.

Draw in diagonals AC and BD. The given information and the shared side AC along with the Reflexive Property can be used to prove ΔABC ≅ ΔADC by the SSS Congruence Postulate. Using CPCTC, ∠B=∠C.The same can be done for ΔABD ≅ ΔBCD using the given information and the shared side BD. This will lead to ∠A=∠C. Therefore, ABCD is a parallelogram because opposite angles are congruent.