In a rhombus PQRS, O is any interior point such that OP = OR. What is the degree measure of ∠ SOQ?

Correct — C.

Every point equidistant from P and R lies on the perpendicular bisector of PR. That is what OP = OR says.

In a rhombus the diagonals bisect each other at right angles, so diagonal QS is exactly the perpendicular bisector of diagonal PR.

So O must sit on line QS, and being interior, it sits between S and Q.

S, O and Q are then collinear, OS and OQ are opposite rays, and ∠SOQ is a straight angle = 180° → option (c).

Why the others are wrong

• (a)
  240° — 240° is larger than a straight angle. With O between S and Q the rays OS and OQ point exactly opposite ways, and both the direct and the reflex reading of that configuration come to 180°.
• (b)
  120° — 120° is a familiar rhombus vertex angle, which is why it reads plausibly. But ∠SOQ is measured at O on the diagonal, not at a vertex, and OP = OR pins O onto SQ.
• (d)
  200° — 200° would need OS and OQ to be non-opposite rays swept the long way round. The condition OP = OR removes that by placing O on segment SQ, where the two rays are opposite.

Two facts meet here, and neither needs a construction.

First, the locus of points equidistant from two fixed points is the perpendicular bisector of the segment joining them. OP = OR therefore confines O to one line, not to a region.

Second, a rhombus has all four sides equal, so its diagonals bisect each other at right angles. Each diagonal is the perpendicular bisector of the other, which means the line OP = OR confines O to is diagonal QS itself.

Once O is on segment SQ, ∠SOQ is a straight angle. The rhombus's shape and O's exact position along QS never enter the answer.

Key facts

• The locus of points equidistant from two fixed points is the perpendicular bisector of the segment joining them.
• A rhombus's diagonals bisect each other at right angles, so each diagonal is the perpendicular bisector of the other.
• OP = OR places O on diagonal QS, making S, O and Q collinear, so ∠SOQ = 180°.
• The answer is independent of the rhombus's angles and of where O sits along QS.

Common traps

• Sketching O somewhere off the diagonal and then measuring an angle from the sketch.
• Confusing OP = OR with OP = OQ, which would put O on the perpendicular bisector of PQ instead.
• Rejecting 180° as 'not a real angle' and choosing the nearest acute or obtuse option.

No directly related past PYQ was found.