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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 : - ICSE Class 10 - Mathematics

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Question

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

Solution

Join DB and DC, IB and IC
∠BAC  66°, ∠ABC  80°, I is the incentre of the ΔABC,
(i) since ∠DBC and ∠DACare in the same segment

∠DBC = ∠DAC
But, `∠DAC = 1 /2 ∠BAC = 1 /2 xx 66° = 33°` =

∴ ∠DBC = 33°

(ii) Since I is the incentre of ∆ABC, IB bisects ∠ABC
`∴ ∠IBC = 1/2 ∠ABC = 1 /2 xx 80° = 40° `

(iii) ∠BAC = 66° and ∠ABC = 80°
In ∠ABC, ∠ACB = 180° - (∠ABC + ∠BAC)
⇒ ∠ACB = 180° - (80° + 66°)
⇒ ∠ACB = 180° - (156°)
⇒ ∠ACB = 34°
Since IC bisects the ∠C

`∴∠ICB = 1/2 ∠C = 1/2xx  34° = 17° `
Now in IBC
∠IBC + ∠ICB + ∠BIC =180°
⇒  40° +17° + ∠BIC = 180°
⇒  57° + ∠BIC = 180°
⇒ ∠BIC = 180° - 57°
⇒ ∠BIC = 123°

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Solution 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 : Concept: Arc and Chord Properties - Angles in the Same Segment of a Circle Are Equal (Without Proof).
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