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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.

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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1) Two opposite angles of a parallelogram are ( 3x – 2)

(50 – x )

Find the measure of each angle of the parallelogram.

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

2nd angle = 37

3rd angle = 180- 37 = 143

4th angle = 143

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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 90^{0} |

4) ∠ABP = ∠CDQ | 4) Alternate interior angles |

5) AB = CD | 5) Properties of parallelogram |

6) ΔAPB = ΔCQD | 6) AAS |

7) AP = CQ | 7) CPCTC |

• 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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