1. Properties of parallelogram
Sides ?
Diagonals ?
Angles ?

2. Theorem Diagonal ? Triangles ?
A diagonal of a parallelogram divides it into two congruent triangles D A B C
Diagonal ?
Triangles ?

3. Proof Given :AC is a diagonal of the parallelogram ABCD
To prove:ΔABC ≡ ΔCDA
Proof :In ΔABC and ΔCDA
∟BCA =∟DAC (Why?)
∟BAC =∟DCA (Why?)
AC = CA (Why?)
ΔABC ≡ ΔCDA (ASA)-proved
The diagonal AC divides parallelogram ABCD into two congruent triangles ABC and CDA A C B D

4. Property -1 (Theorem 8.2) In a parallelogram opposite sides are equalB C
In parallelogram ABCD
AB =DC
BC=AD

5. Property -2 (Theorem 8.4) In a parallelogram opposite angles are equalD A B C
In parallelogram ABCD
∟A = ∟C
∟B = ∟D

6. Property -3 (Theorem 8.6)
The diagonals of a parallelogram bisect each other
The diagonals AC and BD bisect each other at O, then
OA= OC
OB = OD D A B C O

7. Question
What are the conditions for a quadrilateral to become a parallelogram?
A quadrilateral is a parallelogram
If each pair of opposite sides are equal
If each pair of opposite angles are equal
If the diagonals of the quadrilateral bisect each other

8. Another Condition (Theorem 8.8)
A Quadrilateral is a parallelogram if a pair of its opposite sides is equal and parallel
ABCD is a parallelogram if AB = DC and AB II DC Or
( if AD=BC and ABIIBC ) D A B C

9. Conditions ? A quadrilateral is a parallelogram
If each pair of opposite sides are equal
If each pair of opposite angles are equal
If the diagonals of the quadrilateral bisect each other
If a pair of opposite side is equal and parallel
A quadrilateral is a parallelogram

10. Midpoint theorem (Theorem 8.9)
The line segment joining the midpoints of any two sides of a triangle is parallel to the third side A
If E and F are midpoints of sides AB and AC of triangle ABC ,then EF II BC E F C B

11. Converse of midpoint theorem ( 8.10)
The line drawn through the midpoint of one side of a triangle, parallel to another side bisects the third side
If E is the midpoint of AB and EF II BC then F is the mid point of AC (i.e.AF =FC ) A F E B C

12. Summary
A diagonal of a parallelogram divides it into two congruent triangles
In a parallelogram opposite sides are equal.
In a parallelogram opposite angles are equal.
The diagonals of a parallelogram bisect each other.
A Quadrilateral is a parallelogram if a pair of its opposite sides is equal and parallel
The line segment joining the midpoints of any two sides of a triangle is parallel to the third side
The line drawn through the midpoint of one side of a triangle, parallel to another side bisects the third side