Parallelogram :

A quadrilateral inwhich both pairs of opposite sides are parallel is called a parallelogram.

Rhombus:

A quadrilateral in which all four sides are equal and both pairs ofopposite sides are parallel is called a rhombus.

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Given:diagonal AC of parallelogram ABCD bisects ∠A.

∠CAB =∠CAD

To show:

(i) It bisects ∠C also,

(ii) ABCD is a rhombus.

Proof:

i)

In ΔADC and ΔCBA,

AD = CB

(Oppositesides of a ||gm)

DC = BA

(Oppositesides of a ||gm)

AC = CA (Common)

Therefore,

ΔADC ≅ ΔCBA

( by SSS congruence rule.)

∠ACD = ∠CAB ……(i)

(by CPCT)

∠BCA= ∠CAD…..(ii)

(By CPCT)

& ∠CAB = ∠CAD…..(iii)

(Given)

From eq i,ii,iii,

All 4 above angles are equal to each other..

Hence, ∠ACD = ∠BCA

Thus, AC bisects ∠C also.

(ii) ∠ACD = ∠CAD (Proved)

AD = CD (Opposite sides of equal anglesof a triangle are equal)

But, AB= CD & AD= BC

[Opposite sides of a parallelogram]

AB = BC = CD = DA

Hence, ABCD is a rhombus.

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ty bro