Theorem Of Parallelograms

Theorem 7

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

Theorem
If a pair of opposite sides of a quadrilateral is equal and parallel, then it is a parallelogram.
Given: A quadrilateral ABCD in which AB‖DC and AB=DC.
To prove: ABCD is a parallelogram.
Construction: Join AC.

Proof:
Statements
Reasons
In △ABC and △CDA
1. ∠BAC=∠ACD 1. Alt. ∠s, AB‖DC and AC is a transversal.
2. AB=DC
2. Given
3. AC=CA
3. Common
4. △ABC≌△CDA 4. SAS axion of congruency.
5. ∠ACB=∠CAD
 ⇒AD‖BC
Hence, ABCD is a parallelogram.
5. c.p.c.t. Alt, ∠s are equal formed by lines AD, BC and transversal AC.