Theorem Of Parallelograms

Theorem 5

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

Theorem 5
If each pair of opposite angles of a quadrilateral is equal, then it is a parallelogram.
Given: A quadrilateral ABCD in which ∠A=∠C and ∠B=∠D.
To prove: ABCD is a parallelogram.

Proof:
Statements
Reasons
1. ∠A=∠C 1. Given
2. ∠B=∠D 2. Given
3.∠A+∠B=∠C+∠D 3. Adding 1 and 2
4. ∠A+∠B+∠C+∠D= 360°
4. Sum of angles of a quadrilateral.
5. 2$($∠A+∠B$)$=360°
⇒∠A+∠B= 180°
⇒ BC‖AD
5. Using line 3 i.e. Sum of co-interior angles=180°,
formed by lines BC,AD and transversal AB.
Similarly, AB ‖DC
Hence, ABCD is a parallelogram.