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Properties Of Rhombus

Theorem: Every rhombus is a parallelogram in which diagonals bisect the interior angles and also bisect each other at right angles.

Prerequisites:
Rhombus (definition)
Parallelogram (definition)
Alternate angles property (proof)
SSS congruency (proof)
Properties of parallelogram (proof)
Angle on a straight line (proof)

Proof:
Let ABCD be a rhombus with AC being one of its diagonal.

In \triangle ABC and \triangle ADC,

\qquad\quad AB = CD\qquad\qquad\qquad\qquad\qquad\text{(by definition of rhombus)}\\ \text{Also,}\;\;\: BC = AD\qquad\qquad\qquad\qquad\qquad\text{(by definition of rhombus)}\\ \qquad\quad AC = AC\qquad\qquad\qquad\qquad\qquad\:\!\text{(common)}

Hence, \triangle ABC\cong\triangle ADC by SSS rule.

Therefore, by CPCTC, \angle BAC = \angle DCA.
Hence by the alternate angles property, AB \parallel CD.

Similarly, as \angle BCA = \angle DAC, BC \parallel AD.

Therefore, ABCD is a parallelogram (see definition of parallelogram).

Now, let BD be another diagonal of ABCD, which intersects AC at E. Since ABCD is a parallelogram, therefore by the property of a parallelogram, AC and BD bisects each other.


In \triangle AED and \triangle CED,

\qquad\quad AD = CD\qquad\qquad\qquad\qquad\qquad\text{(by definition of rhombus)}\\ \text{Also,}\;\;\: AE = CE\qquad\qquad\qquad\qquad\qquad\:\!\text{(property of parallelogram)}\\ \qquad\quad DE = DE\qquad\qquad\qquad\qquad\qquad\text{(common)}

Hence, \triangle AED\cong\triangle CED by SSS rule.

Therefore, by CPCTC, \angle AED = \angle DEC.

Also, since AC is a straight line,

\therefore\quad\;\;\;\angle AED + \angle DEC = 180^o
\Rightarrow\quad\;\;\angle AED = \angle DEC = 90^o\qquad\qquad\quad\text{(from above)}

Hence the diagonals bisect each other at right angles.

Further, since \triangle AED\cong\triangle CED, therefore by CPCTC,

\qquad\quad\angle ADE = \angle CDE

Hence \angle D is bisected by the diagonal BD. Similarly, all the interior angles are bisected by the diagonals.

Q.E.D.


Recommended:
Characteristics of parallelogram
Angle sum property of polygon
Area of rectangle

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