Question : If O is the orthocenter of a triangle ABC and $\angle$ BOC = 100$^\circ$, then the measure of $\angle$BAC is ____.
Option 1: 100$^\circ$
Option 2: 180$^\circ$
Option 3: 80$^\circ$
Option 4: 200$^\circ$
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Correct Answer: 80$^\circ$
Solution : Given, triangle ABC with O as an orthocentre. And, $\angle$ BOC = 100$^\circ$ So, $\angle$ ODA = $\angle$ OEA = 90$^\circ$ (orthocentre of a triangle) And $\angle$ BOC = $\angle$ DOE = 100$^\circ$ (vertically opposite angles) In quadrilateral ADOE, $\angle$ ADO + $\angle$ DOE + $\angle$ OEA + $\angle$ DAE = 360$^\circ$ Or, 90$^\circ$ +100$^\circ$ + 90$^\circ$ + $\angle$ DAE = 360$^\circ$ Or, $\angle$ DAE = 360$^\circ$ – 280$^\circ$ = 80$^\circ$ Hence, the correct answer is 80$^\circ$.
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