$In the given figure, side BC of Δ A B C has been produced on the left-hand side from B to D and on the right-hand side from C to E. If ∠ A B D = 106^{\circ} a n d ∠ A C E = 118^{\circ}, find the measure of each anlge of the triangle.$

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Solution

Side BC of triangle ABC is produced to D.
∴∠ABD=∠A+∠C⇒106°=∠A+∠C ...(i)
Also, side BC of triangle ABC is produced to E.
∠ACE=∠A+∠B⇒118°=∠A+∠B ...(ii)
Adding (i) and (ii), we get:∠A+∠A+∠B+∠C=106+118°
⇒∠A+∠B+∠C+∠A=224° (∵∠A+∠B+∠C=180°)

⇒180°+∠A=224°⇒∠A=44°
∴∠B=118°-∠A (Using ii)

⇒∠B=118-44°⇒∠B=74°
And,
∠C=106°-∠A (Using i)

⇒∠C=106-44°⇒∠C=62°

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In the given figure, side BC of ΔABChas been produced on the left-hand side from B to D and on the right-hand side from C to E. If ∠ABD=106∘ and ∠ACE=118∘,find the measure of each anlge of the triangle.

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