ABCD is a square . its …

ABCD is a square whose diagonals AC and BD intersect each other at right angles at O. (i)∴∠AOB=∠AOD=90 ∘ In △ANB ∠ANB=180 ∘ −(∠NAB+∠NBA) ⇒∠ANB=180 ∘ −(45 ∘ + 2 45 ∘ ​ ) since NB is bisector of ∠ABD ⇒∠ANB=180 ∘ −45 ∘ − 2 45 ∘ ​ ⇒∠ANB=135 ∘ − 2 45 ∘ ​ But ∠LNO=∠ANB(vertically opposite angles) ∴∠LNO=135 ∘ − 2 45 ∘ ​ ........(1) Now in △AMO, ∠AMO=180 ∘ −(∠AOM+∠OAM) ⇒∠AMO=180 ∘ −(90 ∘ + 2 45 ∘ ​ ) since MA is bisector of ∠DAO ⇒∠AMO=180 ∘ −90 ∘ − 2 45 ∘ ​ ⇒∠AMO=90 ∘ − 2 45 ∘ ​ ........(2) Adding (1) and (2) we get ∠LNO+∠AMO=135 ∘ − 2 45 ∘ ​ +90 ∘ − 2 45 ∘ ​ ⇒∠LNO+∠AMO=225 ∘ −45 ∘ =180 ∘ ⇒∠LNO+∠AMO=180 ∘ (ii)∠BAM=∠BAO+∠OAM ⇒∠BAM=45 ∘ + 2 45 ∘ ​ =67 2 1 ​ ∘ And ⇒∠BMA=180 ∘ −(∠AOM+∠OAM) ⇒∠BMA=180 ∘ −(90 ∘ + 2 45 ∘ ​ ) ⇒∠BMA=90 ∘ − 2 45 ∘ ​ =67 2 1 ​ ∘ ∴∠BMA=∠BAM (iii) In quadrilateral ALOB ∵∠ABO+∠ALO=45 ∘ +90 ∘ +45 ∘ =180 ∘ ∴,ALOB is a cyclic quadrilateral.