Download Properties of parallelogram

Properties of parallelogram
Sides ?
Diagonals ?
Angles ?
Theorem
 A diagonal of a parallelogram divides it into two
congruent triangles
C
D
Diagonal ?
A
B
Triangles ?
Proof
 Given :AC is a diagonal of the parallelogram ABCD
 To prove:ΔABC ≡ ΔCDA
 Proof :In ΔABC and ΔCDA
D
∟BCA =∟DAC (Why?)
∟BAC =∟DCA (Why?)
AC = CA (Why?)
A
B
ΔABC ≡ ΔCDA (ASA)-proved
The diagonal AC divides parallelogram ABCD into two
congruent triangles ABC and CDA
C
Property -1 (Theorem 8.2)
 In a parallelogram
opposite sides are equal
C
D
 In parallelogram ABCD
 AB =DC
A
B
 BC=AD
Property -2 (Theorem 8.4)
 In a parallelogram
opposite angles are
equal
C
D
 In parallelogram ABCD
 ∟A = ∟C
 ∟B = ∟D
A
B
Property -3 (Theorem 8.6)
 The diagonals of a
parallelogram bisect
each other
 The diagonals AC and BD
C
D
O
A
B
bisect each other at O,
then
 OA= OC
 OB = OD
Question
 What are the conditions
for a quadrilateral to
become a parallelogram?
 A quadrilateral is a
parallelogram
1) If each pair of opposite
sides are equal
2) If each pair of opposite
angles are equal
3) 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
C
D
A
B
 ABCD is a parallelogram
if AB = DC and AB II DC
 Or
( if AD=BC and ABIIBC )
Conditions ?
 A quadrilateral is a
parallelogram
1) If each pair of opposite
sides are equal
2) If each pair of opposite
angles are equal
3) If the diagonals of the
quadrilateral bisect
each other
4) If a pair of opposite
side is equal and
parallel
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
E
B
sides AB and AC of triangle
ABC ,then EF II BC
F
C
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
A
E
B
 If E is the midpoint of
AB and EF II BC then F is
the mid point of AC
(i.e.AF =FC )
F
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