Circle angle and arc relationships: which statements are correct?

Key takeaway: These problems usually test whether you know the three classic circle facts: central angles match arcs, angles on a radius triangle can be found from equal sides, and arc measures depend on the angle type.

Let’s test each statement one by one.

Given

• Circle $\odot M$
• $m\angle LMN=110^\circ$

a. $m\overset{\frown}{LN}=110^\circ$

• This is true.
• A central angle has the same measure as its intercepted arc.

b. $m\angle OLN=65^\circ$

• This is not enough information as written unless the diagram shows $MO=OL$ or some other specific triangle relation.
• If the figure implies $\triangle OLN$ is isosceles with vertex angle $50^\circ$, then each base angle would be $65^\circ$.
• So this statement can only be judged from the full diagram.

c. $m\overset{\frown}{OL}=120^\circ$

• This may be true or false depending on the diagram’s marked angles/arcs.
• On its own, the given information does not force $120^\circ$.

What to remember

• Central angle = intercepted arc.
• Base angles of an isosceles triangle are equal.
• Don’t guess missing arc measures unless the diagram shows the needed markings.

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Pitfalls the pros know 👇 Students often treat every angle in a circle as a central angle. It’s safer to check the vertex first. Another trap is assuming two radii are equal when the diagram never marks them. If the proof depends on an isosceles triangle, you need that side information visible.

What if the problem changes? If the question had asked only for the arc intercepted by $\angle LMN$, the answer would be $110^\circ$. If the figure showed $MO=OL$, then $\triangle MOL$ would be isosceles and you could use equal base angles to find the missing angle at $L$.

Tags: central angle, inscribed angle, isosceles triangle