LEARN MATH STEP BY STEP THROUGH VERY EASY PROCESS

THEOREM OF PARALLELOGRAM

**Parallelogram –**

**A quadrilateral will be considered as a parallelogram when its opposite sides are parallel. In the figure, ABCD is a parallelogram, in which DC is parallel to AB & DA is parallel to CB. A parallelogram has some special properties which we learn –**

**Theorem.1)**

**1) The opposite sides of a parallelogram are to be considered equal.**

**2) The opposite angles of a parallelogram are to be considered equal.**

**3) Each diagonal bisects a parallelogram into two congruent triangles.**

**Given
– ABCD is a parallelogram in which BC || AD and BA || CD . **

**To Prove – 1) BC = AD and BA = CD**

**2) ∠A
= ∠C and ∠B = ∠D**

**
3) ∆ BCD ≅ ∆ DAB and ∆ BCA
≅ ∆ DAC**

**Construction – Join the point B & D **

**Proof -
in ∆ BCD and ∆ DAB**

** ∠1 = ∠2 (BC || AD, alternate
angles are equal),**

** BD = BD (Common)**

**
And, ∠3 = ∠4 (BA || CD, alternate
angles are equal), **

**So,
∆ BCD ≅ ∆ DAB (A-S-A condition of congruency),**

**So, the corresponding parts of the
triangles are equal**

**So, BC = AD and BA = CD (Proved)**

**Also, ∠A = ∠C**

**We have, ∠1 = ∠ 2
and ∠4 = ∠3**

**So, ∠1 + ∠ 4
= ∠2 + ∠3 **

**=> ∠B = ∠D**

**So, ∠A = ∠ C
and ∠B = ∠ D (proved)**

**Now, ∆ BCD ≅ ∆ DAB (Proved already),**

**Similarly, ∆ BCA ≅ ∆ DAC**

**Hence, each diagonal bisects the
parallelogram into two congruent parts.
(proved)**

**Theorem.2)****The diagonals of a parallelogram bisect each other.**

**
**

**Given - ABCD is a parallelogram in which AB || DC, AD || BC and the diagonals AC & BD intersect at the point O.**

**To prove – OA = OC, and OB = OD**

**Proof - ∆ OAB and ∆ OCD**

**∠OAB = ∠OCD (AB || DC and alternate angles are equal)**

** AB = DC (Opposite sides of a parallelogram
are equal)**

** And, ∠OBA = ∠ODC
(AB || DC and alternate angles
are equal)**

** ∆ OAB ≅ ∆ OCD (A-S-A condition of congruency)**

**So, the corresponding sides of ∆ OAB and ∆ OCD are equal**

**So, OA = OC and OB = OD (proved)**

**Theorem.3)****If
a pair of opposite sides of a quadrilateral are equal and parallel, the
quadrilateral is a parallelogram.**

**Ans.) Given – ABCD is a
quadrilateral in which AB = DC and AB || DC.**

**To prove – ABCD is a parallelogram**

**Construction – Join the point B & D**

**Proof – In ∆ ABD and ∆ CDB,**

**
AB = DC (Given)**

** ∠ABD = ∠CDB (AB || DC and alternate angles are equal)**

** BD = DB (common side)**

** ∆ ABD ≅ ∆ CDB (S-A-S condition of congruency)**

**So, it can be concluded the
corresponding parts of these triangles are equal**

**So, ∠ADB = ∠CBD, but
these are alternate angles, so AD || BC.**

** Thus, AB || DC and AD || BC.**

**Hence, ABCD is a parallelogram.**