Parallelogram
Definition: A quadrilateral having its opposite sides
parallel is called a parallelogram. In the adjoining figure, AB∥CD and AD∥BC. So ABCD is a
parallelogram.
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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, AB∥DC and BC∥AD.
To prove: AB=CD and AD=BC
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, AB∥DC and BC∥AD.
To prove: ∠BAD = ∠BCD and ∠ABC = ∠ADC
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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parallelogram here.
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