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
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

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