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In \( \Large \triangle ABC \), AB = AC and D is a point on AB, such that AD = DC = BC. Then \( \Large \angle BAC \), is


A) \( \Large 40 ^{\circ} \)

B) \( \Large 45 ^{\circ} \)

C) \( \Large 30 ^{\circ} \)

D) \( \Large 36 ^{\circ} \)

Correct Answer:
D) \( \Large 36 ^{\circ} \)

Description for Correct answer:

Given that AB = AC and AD = CD = BC

\( \Large \angle ABC = \theta \)

Then, \( \Large \angle ACB = \theta \) [\( \Large \because AB = AC \)]

=> \( \Large \angle BAC = 180 ^{\circ} - 20 \)

=> \( \Large \angle ACD = 180 - 20 \) [\( \Large \because AD = CD \)]

\( \Large \angle BCD = \angle AC B - \angle ACD \)

=> \( \Large \angle BCD = \theta - \left(180 ^{\circ} - 20 \right) = 30 - 180 ^{\circ} \)

and \( \Large \angle BDC = \theta \) [\( \Large \because CD = BC \)]

Now, in \( \Large \triangle BCD \)

\( \Large \angle CBD + \angle BDC + \angle BCD = 180 ^{\circ} \)

=> \( \Large \theta + \theta + 30 - 180 ^{\circ} = 180 ^{\circ} \)

=> \( \Large 50 = 360 ^{\circ} => \theta = 72 ^{\circ} \)

\( \Large \therefore \angle BAC = 180 ^{\circ} - 20 \)

= \( \Large 180 ^{\circ} - 144 = 36 ^{\circ} \)


Part of solved Geometry questions and answers : >> Elementary Mathematics >> Geometry




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