Find the angle X

Geometry Level 3


The answer is 30.

Aly Ahmed by Aly Ahmed , 34, Egypt

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1 solution

Let ∠ A D C = α \angle {ADC}=α and ∠ C B D = β \angle {CBD}=β . Then 2 α = 180 ° − β + 6 ° = 186 ° − β 2α=180\degree-β+6\degree=186\degree-β . Also, 180 ° − β = 180 ° − α + 12 ° 180\degree-β=180\degree-α+12\degree or β = α − 12 ° β=α-12\degree . So, 2 α = 186 ° − α + 12 ° 2α=186\degree-α+12\degree or 3 α = 198 ° 3α=198\degree or α = 66 ° α=66\degree . Therefore β = 66 ° − 12 ° = 54 ° β=66\degree-12\degree=54\degree . So, ∠ A D C = 66 ° , ∠ A B C = 180 ° − 2 β = 180 ° − 108 ° = 72 ° \angle {ADC}=66\degree, \angle {ABC}=180\degree-2β=180\degree-108\degree=72\degree . So ∠ D A C = 180 ° − 144 ° − 6 ° = 30 ° \angle {DAC}=180\degree-144\degree-6\degree=\boxed {30\degree}

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Why is ∠ A D C \angle ADC equal to 6 6 ∘ 66^\circ ? In ∠ A B C = 18 0 ∘ − 10 8 ∘ \angle ABC = 180^\circ - 108^\circ , where does 10 8 ∘ 108^\circ come from?

Jon Haussmann - 1 year, 4 months ago

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I have elaborated the answer to your queries.

A Former Brilliant Member - 1 year, 4 months ago

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