# Kite and its Theorems

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In this section, we will discuss kite and its theorems.
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