Question

If AD and BD are bisectors of ∠CAB and ∠CBA, respectively. Show that ∠ADB=90°+12∠ACB.

Answer 1

Given :  AD and BD are bisectors of ∠CAB and ∠CBA  

To find :  Show that ∠ADB. = 90° +  (1/2)   ∠ACB

Solution:

in ΔABC

∠CAB + ∠CBA   + ∠ACB. = 180°    Sum of angles of triangle

=>  ∠CAB + ∠CBA   =  180° - ∠ACB.

in ΔDBC

∠DAB + ∠DBA   + ∠ADB. = 180°    Sum of angles of triangle

AD and BD are bisectors of ∠CAB and ∠CBA,

∠DAB = (1/2)  ∠CAB

∠DBA = (1/2)  ∠CBA

=> (1/2)  ∠CAB  + (1/2)  ∠CBA    + ∠ADB. = 180°  

=> (1/2)   (  ∠CAB + ∠CBA  ) + ∠ADB. = 180°  

=>  (1/2)   (  180° - ∠ACB. ) + ∠ADB. = 180°  

=> 90° -  (1/2)   ∠ACB + ∠ADB. = 180°  

=>  ∠ADB. = 90° +  (1/2)   ∠ACB

QED

Hence Proved

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