if i is the incentre of triangle abc – RNCREATE

questions

if i is the incentre of triangle abc

If I is the incentre of triangle ABC and AI when produced meets the circumcircle of triangle ABC in point D.If ∠BAC=66∘ and ∠ABC=80∘

Calculate:(i) ∠DBC (ii) ∠IBC, (iii) ∠BIC.

. solutions

Given ∠BAC=66∘ and ∠ABC=80∘(i) ∠DBC=∠DAC (angles in the same segment)Since I is the incenter. The incentre of the triangle is the point of intersection of the internal bisector of angles.

if i is the incentre of triangle abc
if i is the incentre of triangle abc

∠DAC=12∠BAC=12×66∘=33∘

Therefore ∠DBC=33∘

(ii) Similarly, ∠IBC=12∠ABC=12×80∘=40∘

(iii) In △ABC,∠ACB=180−∠ABC−∠BAC=180−80−66=34∘Also ∠ICB=12∠BCA=12×34∘=17∘ie. ∠ICB=12×34∘=17∘∠BIC=180∘−∠IBC−∠ICB=180∘−40∘−17∘=123∘

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