The exterior angles, obtained on producing the base of a triangle(side $B C$ ) both ways, are $102^{\circ}$ and $134^{\circ}$ . Find the $∠ A$ (in degrees)

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Solution

Let the triangle be ABC, with $102^{\circ}$ formed by extending along B and $134^{\circ}$ be formed by extending along C
Thus, $∠ A + ∠ C = 102^{\circ}$ (Exterior angle is equal to sum of interior opposite angles)

$∠ A + ∠ B = 134^{\circ}$ (Exterior angle is equal to sum of interior opposite angles)
Adding the two equations,
$∠ A + ∠ A + ∠ B + ∠ C = 236^{\circ}$
we know, $∠ A + ∠ B + ∠ C = 180^{\circ}$ (Sum of angles of a triangle)
Hence, $∠ A = 236^{\circ} - 180^{\circ}$
$∠ A = 56^{\circ}$

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