Question

In the figure given below, LM = LN; angle PLN = 110^o.

calculate: (i) ∠LMN
                 (ii) ∠MLN

Answer 1

Given: ∠PLN = 110°
(i) We know that the sum of the measure of all the angles of a quadrilateral is 360°. 
In quad. PQNL,
∠QPL + ∠PLN + ∠LNQ + ∠NQP = 360°
⇒ 90° + 110° + ∠LNQ + 90° = 360°
⇒ ∠LNQ = 360° − 290°
⇒ ∠LNQ = 70°
⇒ ∠LNM = 70° ........(i)
In ΔLMN,
LM = LN ........( Given )
∴ ∠LNM = ∠LMN ....... [angles opp. to equal sides are equal]
⇒ ∠LMN = 70° ....(ii) [ from(i) ]

(ii) In ΔLMN,
∠LMN + ∠LNM+ ∠MLN = 180°
But ∠LNM= ∠LMN = 70° .....[ From(i) and (ii)]
∴ 70° + 70° + MLN = 180°
⇒ ∠MLN = 180°− 140°
⇒ ∠MLN = 40°