# Kite and its Theorems

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In kite, adjacent sides are equal and long diagonal bisect the small diagonal at right angle.All interior angles are acute angles.

**Theorem 1 : If a quadrilateral is a kite, then its diagonals are perpendicular.**

**GIVEN :**AB ≅ CB and AD ≅ CD

**PROVE THAT :**AC ⊥ BD

Proof :

Statements |
Reasons |

1)AB ≅ AD | 1) Given |

2) BC ≅ CD | 2) Given |

3) AC ≅ AC | 3) Reflexive (common side) |

4) ΔABC ≅ ΔADC | 4) SSS Postulates |

5) ∠BAE ≅ ∠DAE | 5) CPCTC |

6) ΔABD is an Isosceles triangle. | 6) By property of an isosceles triangle. |

7) ∠ABE ≅ ∠ADE | 7) Property of isosceles triangle. |

8) ΔABE ≅ ΔADE | 8) ASA postulate. |

9) ∠AEB ≅ ∠AED | 9) CPCTC |

10)∠AEB +∠AED = 180 | 10) Linear pair angles are supplementary. |

11) 2∠AEB = 180 | 11) Addition property |

12) ∠AEB = 90 | 12) Division property |

13) AC ⊥ BD and AE ⊥ BD |
13) By property of perpendicular. |

**Theorem 2: If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent.**

**Given :**ABCD is a kite with AB ≅ AD and BC ≅ CD.

**Prove that :**∠A ≅ ∠C.

Proof :

Statements |
Reasons |

1)AB ≅ AD | 1) Given |

2) BC ≅ CD | 2) Given |

3) AC ≅ AC | 3) Reflexive (common side) |

4) ΔABC ≅ ΔADC | 4) SSS Postulates |

5) ∠ABC ≅ ∠ADC | 5) CPCTC |

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**Example based on kite and its theorems :**

In a kite, ABCD,AB = x + 2 , BC = 2x + 1. The perimeter of kite is 48cm. Find x and also find the length of each side.

**Solution :**

As we know that, adjacent sides in a kite are equal.

∴ AB = AD and BC = CD.

Perimeter = sum of all the sides

P = AB + BC + CD + AD

48 = x + 2 + 2x + 1 + x + 2 + 2x + 1

48 = 6x + 6

⇒ 6x = 48 -6

∴ 6x = 42

x = 42/6

x = 7

∴ AB = AD = x + 2 = 7 + 2 = 9cm

and BC = CD = 2x + 1 = 2(7) + 1 = 14 + 1 = 15 cm

**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

• 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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