A geometry problem by Satyabrata Dash

Geometry Level pending

Let A B C ABC be a triangle with ∠ B A C = 117 ◦ ∠BAC = 117◦ . The angle bisector of ∠ A B C ∠ABC intersects side A C AC at D D . Suppose ▵ A B D ∼ ▵ A C B ▵ABD ∼ ▵ACB . Compute the measure of ∠ A B C ∠ABC , in degrees.

10.5 21 84 117 42 58.5

Satyabrata Dash by Satyabrata Dash , 20, India

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1 solution

Michael Mendrin
May 31, 2018

Let x x be angle ∠ A B D \angle ABD .

Then ∠ D A C = ∠ A C B = x \angle DAC = \angle ACB = x , so that 180 = 117 + 3 x 180 = 117 + 3x , or x = 21 x=21 .

Hence ∠ A B C = 42 \angle ABC = 42

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