Properties of parallelogram Sides ? Diagonals ? Angles ?

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

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

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

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

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

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

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

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

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

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

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