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      Question

      In a rhombus ABCD, O is any interior point such that OA

      = OC. What is five-ninths of the degree measure of ∠DOB?
      A 120° Correct Answer Incorrect Answer
      B 150° Correct Answer Incorrect Answer
      C 100° Correct Answer Incorrect Answer
      D 90° Correct Answer Incorrect Answer

      Solution

      ABCD is a rhombus. O is an interior point such that OA = OC. We know: In a rhombus, diagonals bisect each other at 90°. So, ∠AOC = ∠BOD = 90°. But here, OA = OC, which means point O lies on the perpendicular bisector of AC, i.e., O lies on diagonal BD. So now, in triangle DOB: Since O lies on BD (diagonal), and OA = OC, that forces O to be on BD, making triangle DOB an isosceles right triangle. Then, ∠DOB = 180°, because now O lies on diagonal BD, and DOB becomes a straight angle. Now (5/9) × 180° = 100°

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