# Parallelogram and its Theorems

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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 = 370
2nd angle = 370

3rd angle = 180- 37 = 1430

4th angle = 1430

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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