# Parallelogram ## Parallelogram

Definition: A quadrilateral having its opposite sides parallel is called a parallelogram. In the adjoining figure, ABCD and ADBC. So ABCD is a parallelogram.

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Square, Rectangle, Rhombus are some special types of parallelograms
Properties of parallelogram
1.       Opposite sides of a parallelogram are equal
2.       Opposite angles of a parallelogram are equal.
3.       Diagonals of a parallelogram bisect each other.

### Proofs:

1.       Prove theoretically that the opposite sides of a parallelogram are equal.
Given: ABCD is a parallelogram where, ABDC and BCAD.
Construction: B and D joined.
Proof:
Statements                          Reasons
1. In ΔABD and ΔBCD
i.    ABD = BDC (A) ---------> Alternate angles
ii.   BD = BD (S) -----------------> Common side
iii. ADB = CBD (A) ---------> Alternate angles
2. ΔABD ΔBCD -------------------> By A.S.A. axiom
3. AD = CD and AD = BC ---------> Corresponding sides of congruent triangles
Proved.

2.       Prove theoretically that the opposite angles of a parallelogram are equal.
Given: ABCD is a parallelogram where, ABDC and BCAD.
Construction: B and D joined.
Proof:
Statements                             Reasons
1.       In ΔABD and ΔBCD
i.    ABD = BDC (A) -------> Alternate angles
ii.   BD = BD (S) ---------------> Common side
iii. ADB = CBD (A) -------> Alternate angles
2.       ΔABD ΔBCD -----------------> By A.S.A. axiom
3.       BAD = BCD ------------------> Corresponding angles of congruent triangles
4.       ABC = ADC ------------------> Similarly by joining A and C
Proved.

3.       Prove theoretically that the diagonals of a parallelogram bisect each other.
Given: ABCD is a parallelogram. Diagonals AC and BD intersect at O.
To prove: AO=CO and BO=DO
Proof:
Statements                                    Reasons
1.       In ΔAOD and ΔBOC
i.   OAD = OCB (A) ------------> Alternate angles
ii.   AD = BC (S) --------------------> Opposite sides of a parallelogram
iii. ODA = OBC (A) ------------> Alternate angles
2.       ΔABD ΔBCD ----------------------> By A.S.A. axiom
3.       AO = CO and BO = DO -----------> Corresponding sides of congruent triangles
Proved.

### Workout Examples

Example 1: Find the values of unknown angles in the given parallelogram.
Solution: From the figure,
a + 120° = 180° ----------------> Co-interior angles.
or,          a = 180° - 120°
or,          a = 60°
b = 120° --------------------------> Opposite angles of a parallelogram.
c = a -----------------------------> Opposite angles of a parallelogram.
= 60°

a = 60°
b = 120°
c = 60°

Example 2: Find the values of unknown angles in the given parallelogram.
Solution: From the figure,
4a + 5a = 180° ----------------> Co-interior angles.
or,         9a = 180°
or,           a = 180°/9
or,           a = 20°
b = 4a --------------------> Corresponding angles.
= 4 × 20°
= 80°
c = b --------------------> Alternate angles.
= 80°
d = 5a --------------------> Opposite angles of a parallelogram.
= 5 × 20°
= 100°

a = 20°
b = 80°
c = 80°
d = 100°

Example 3: Find the values of unknown angles in the given parallelogram.
Solution: From the figure,
a + 35° + 60° = 180° ----------------> Sum of angles of ΔABC.
or,          a + 95° = 180°
or,          a = 180° - 95°
or,          a = 85°
b = 35° --------------------------> Alternate angles.
c = a -----------------------------> Alternate angles.
= 85°
d = 60° --------------------------> Opposite angles of a parallelogram.
a = 85°
b = 35°
c = 85°
d = 60°

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