Parallelogram and its Theorems

In this section we will discuss parallelogram and its theorems.
On the basis of properties of parallelogram there are different theorems.

Examples

1) A diagonal of a parallelogram divides it into two congruent triangles.
Given : Δ ABCD is a parallelogram and AC is a diagonal.
Prove that : ΔABC = Δ CDA

Statements Reasons
ABCD is a parallelogram Given
AC is a diagonal given Given
∠BCA = ∠DAC Alternate interior angles
∠BAC = ∠DCA Alternate interior angles
AC = CA Reflexive (common side)
ΔABC = ΔCDA ASA postulate

So diagonal AC divides parallelogram ABCD into two congruent triangles ABC and CDA.

Statements of parallelogram and its theorems
1) In a parallelogram, opposite sides are equal.
2) If each pair of opposite sides of a quadrilateral is equal then it is a parallelogram.
3) In a parallelogram, opposite angles are equal.
4) If in a quadrilateral, each pair of opposite angles is equal then it is a parallelogram.
5) The diagonals of a parallelogram bisect each other.
6) If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.
7) A quadrilateral is a parallelogram if a pair of opposite sides is equal and parallel.

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Some solved examples using parallelogram and its theorems

1) Two opposite angles of a parallelogram are ( 3x – 2)
0 and
(50 – x )
0 .
Find the measure of each angle of the parallelogram.

Solution :
Opposite angles of parallelogram are equal.

3x – 2 = 50 – x

⇒ 3x + x = 50 + 2

⇒ 4x = 52

∴ x = 13

1st angle = 3x – 2 = 3(13) – 2 = 37
0
2nd angle = 37
0

3rd angle = 180- 37 = 143
0

4th angle = 143
0

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2) In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ.

Given : ABCD is a parallelogram. DP = BQ.

Prove that 1) ΔAPD = ΔCQB
2) AP = CQ 3) Δ AQB = ΔCPD 4) AQ =CP 5) APCQ is a parallelogram.

Statements Reasons
1) ABCD is a parallelogram 1) Given
2) DP = BQ 2) Given
3) AD = BC 3) Properties of parallelogram
4) ∠ADP = ∠CBQ 4) Alternate interior angles
5) ΔAPD =ΔCQB 5) SAS Postulate
6) AP = CQ 6) CPCTC
7) AB = CD 7) Properties of parallelogram
8) BQ = DP 8) Given
9) ∠ABQ = ∠CDP 9) Alternate interior angles
10) ΔAQB = ΔCPD 10) SAS Postulate
11) AQ = PC 11) CPCTC
12) APCQ is a parallelogram 12) Two pairs of opposite sides of ≅then it is parallelogram. From (6) and (11)

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3) ABCD is a parallelogram and AP and CQ are perpendiculars from A and C on diagonal BD.Prove that AP = CQ.


Statements Reasons
1) ABCD is a parallelogram 1) Given
2) AP and CQ are perpendiculars 2) Given
3) ∠APB = ∠CQD 3) Definition of perpendiculars.Each of measure 900
4) ∠ABP = ∠CDQ 4) Alternate interior angles
5) AB = CD 5) Properties of parallelogram
6) ΔAPB = ΔCQD 6) AAS
7) AP = CQ 7) CPCTC

Quadrilateral

Introduction to Quadrilateral
Types of Quadrilateral
Properties of Quadrilateral
Parallelogram and its Theorems
Rectangle and its Theorems
Square and its Theorems
Rhombus and its Theorems
Trapezoid (Trapezium)and its Theorems
Kite and its Theorems
Mid Point Theorem
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