In a circle, if points $P$, $Q$, and $R$ lie on the circle with center $O$, and if $\angle PQR = 100^\circ$, we can find $\angle POR$ using the property that the angle subtended at the center by an arc is twice the angle subtended at any point on the circumference.

Given: $\angle PQR = 100^\circ$

Since $\angle POR$ is the angle at the center subtended by the same arc $PR$, it will be twice $\angle PQR$. Therefore: $\angle POR = 2 \times \angle PQR = 2 \times 100^\circ = 200^\circ$

Thus, $\angle POR = 200^\circ$

Would you like further details or explanations on any related concepts?

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Here are five related questions to expand your understanding:

1. What is the relationship between angles at the center and on the circumference for the same arc in any circle?
2. How do inscribed angles in the same circle compare if they subtend the same arc?
3. If $\angle PQR$ were different, how would it affect $\angle POR$?
4. Can you explain why the angle at the center is always twice the angle at the circumference for the same arc?
5. What would happen if $O$ were not the center of the circle?

Tip: In circle geometry, remember that the angle subtended by an arc at the center is always twice the angle subtended by the same arc at any point on the circle's circumference.