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Question

# In the given figure, O is a point in the interior of 1 M a square ABCD such that OAB is an equilateral triangle. Show that OCD is an isosceles triangle.

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Solution

## We know that △OAB is an equilateral triangleSo it can be written as∠OAB=∠OBA=AOB=60∘From the figure we know that ABCD is a squareSo we get∠A=∠B=∠C=∠D=90∘In order to find the value of ∠DAOWe can write it as∠A=∠DAO+∠OABBy substituting the values we get90∘=∠DAO+60∘On further calculation∠DAO=90∘−60∘By subtraction∠DAO=30∘We also know that ∠CBO=30∘Considering the △OAD and △OBCWe know that the sides of a square are equalAD=BCWe know that the sides of an equilateral triangle are equalOA=OBBy SAS congruence criterion△OAD≅△OBCSo we get OD=OC(c.p.c.t)Therefore, it is proved that △OCD is an isosceles triangle.

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