Question

In a rhombus KLMN, the diagonal KM is equal to the side of the rhombus. Find the measure of angleLMN.
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Answer 1

Question :- In a rhombus KLMN, the diagonal KM is equal to the side of the rhombus. Find the measure of ∠LMN. ?

Solution :-

Given that,

→ KLMN is a rhombus.

So,

→ KL = LM = MN = NL . (All sides of rhombus are equal in length .)

Also,

→ Diagonal KM = sides of the rhombus .

Therefore,

→ KM = KL = LM = MN = NL .

So, From image, we have ,

→ ∆ KNM and, ∆KLM have all sides equal.

Hence,

∆ KNM and, ∆KLM both are Equilateral triangle's whose all sides are equal in length and each angle is equal to 60° .

Therefore,

→ ∠LMN = ∠LMK + ∠KMN

→ ∠LMN = 60° + 60°

→ ∠LMN = 120° .(Ans.)

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Answer 2

Given KLMN is a Rhombus and Diagonal KM is equal to the sides of the Rhombus.

We know that,

KL = LM = MN = NK ( All sides are equal )

$\pink { KM} = \blue{KL = LM = MN = NK}\: ( given )$

$In \: \triangle KLM , KL = LM = KM$

$\therefore \triangle KLM \:is \: equilateral \:triangle$

$\angle {MKL} = \angle {KLM} = \angle {LMK} = 60\degree$

$Similarly , \angle {KMN} = 60\degree$

$\angle { LMN } = \angle {LMK} + \angle {KMN}\\= 60\degree + 60\degree \\= 120\degree$

Therefore.,

$\red{\angle { LMN }}\green { = 120\degree}$

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