In the figure given below, LM = LN; angle PLN = 110°. Calculate :

(i) ∠LMN

(ii) ∠MLN

(i) In quadrilateral PQNL,

By angle sum property of quadrilateral,

⇒ ∠QPL + ∠PLN + ∠LNQ + ∠NQP = 360°

⇒ 90° + 110° + ∠LNQ + 90° = 360°

⇒ ∠LNQ + 290° = 360°

⇒ ∠LNQ = 360° - 290° = 70°.

From figure,

⇒ ∠LNM = ∠LNQ = 70°.

In triangle LMN,

⇒ LN = LM (Given)

⇒ ∠LMN = ∠LNM = 70°.

Hence, ∠LMN = 70°.

(ii) In triangle LMN,

By angle sum property of triangle,

⇒ ∠LMN + ∠LNM + ∠MLN = 180°

⇒ 70° + 70° + ∠MLN = 180°

⇒ ∠MLN + 140° = 180°

⇒ ∠MLN = 180° - 140° = 40°.

Hence, ∠MLN = 40°.