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# Properties of Rhombus - Property: The diagonals of a rhombus are perpendicular bisectors of one another.

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# The diagonals of a rhombus are perpendicular bisectors of one another. Given: ABCD is a rhombus.

To Prove: m∠ AOD = m∠ COD = 90°.

Proof:

ABCD is a rhombus.                                                                         .........(Given)

Since the opposite sides of a rhombus have the same length, it is also a parallelogram.  ........(Properties of a rhombus)

The diagonals of a rhombus bisect each other.                            ..........(Properties of a rhombus)

Thus,

OA = OC            .....(DB ⟂ AC, Divides AO and OC into two equal parts)(1)

OB = OD            ......(AC ⟂ DB, Divides DO and OB into two equal parts)(2)

In ∆AOD and ∆COD,
OA = OC             .....(From 1)
OB = OD             ......(From 2)
AD = CD             ......(All the sides of a rhombus are equal.)

by SSS congruency criterion,
∆ AOD ≅ ∆ COD

Therefore, m∠ AOD = m ∠ COD.......(C.A.C.T.)

Since, ∠AOD and ∠ COD are a linear pair.

m∠ AOD = m∠ COD = 90°.

Hence Proved.

#### Example

RICE is a rhombus. Find x, y, z. Justify your findings. x = OE = OI = 5       .....(diagonals bisect)
y = OR = OC = 12.  .....(diagonals bisect)
z = side of the rhombus = 13 .......(all sides are equal).
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To Prove that the Diagonals of a Rhombus Bisect Each Other at Right Angles [00:14:21]
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##### Series: Property: the Diagonals of a Rhombus Are Perpendicular Bisectors of One Another.
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