A circle relationship requires the same arc
A supplied circle model identifies O as its center and P,Q,R as points on the circle. The central angle POQ measures 84 degrees. An inscribed angle PRQ is explicitly stated to intercept the same arc PQ as that central angle, with the appropriate vertex on the circle. A learner doubles 84 instead of halving it. Another halves an unrelated angle drawn nearby. A third names the theorem’s conditions before calculating. The task asks for the inscribed angle and what must be checked before using the relationship. No unseen diagram is required because the relevant center, vertices and intercepted arc are specified in words. The model does not give every other arc or angle in the circle. A correct response should apply the stated relationship without treating any pair of circular-looking angles as interchangeable.
1. What is the inscribed angle?
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It is half the central angle intercepting the same arc:84/2=42 degrees. The stated center, on-circle vertex and shared arc justify the relationship.
2. What changes would invalidate this shortcut?
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If the angles intercepted different arcs, or the supposed central vertex were not at the center, the stated central-inscribed relationship could not simply be applied unchanged.