Theorem Of Parallelograms

Theorem 3

If each pair of opposite sides of a quadrilateral is equal, then it is a parallelogram.

Theorem 3
If each pair of opposite sides of a quadrilateral is equal, then it is a parallelogram.
Given. A quadrilateral ABCD in which AB = DC and BC = AD.
Construction. Join AC.

Proof:

Statements
Reasons
In△ABC and △CDA
1. AB=DC
1. Given.
2. BC=AD
2. Given.
3. AC=AC
3. Common.
4. △ABC ≌ △ CDA 4. SSS Congruency Rule
5. ∠ BAC= ∠ ACD ⇒ AB ∥ DC 5. C.P.C.T alt. ∠s are equal formed by lines AB,DC and transversal AC.
6. ∠ ACB=∠ CAD ⇒ BC∥AD 6. c.p.c.t. alt, ∠s are equal formed by lines AD, BC and transversal AC.

Hence, ABCD is a parallelogram.▆