Which circle geometry statements are correct after an angle is given?

Expert intro: These problems reward careful reading. One correct central angle can unlock arc measures, but only if the statement matches the exact transformation in the diagram.

Detailed walkthrough

We are given $\odot B$ with $m\angle ABC=130^\circ$.

Check each statement

a. $m\overset{\frown}{AC}=130^\circ$

• True if $\angle ABC$ is a central angle.
• Central angle measure equals intercepted arc measure.

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

• This is possible only if the diagram shows $\triangle CAD$ is isosceles and $\angle ACD=65^\circ$ or similar structure.
• The given information alone does not force this.

c. $m\overset{\frown}{DA}=100^\circ$

• Not determined from the given angle alone.
• You would need more arc or angle information.

Best conclusion

• The only statement directly supported by the standard circle rule is a.

Why

• Central angle = intercepted arc.
• The other statements require extra diagram markings or triangle relationships.

💡 Pitfall guide

A frequent mistake is to assume every labeled point on the circle creates a known arc. It doesn’t. You can only claim an arc measure when the diagram or a theorem gives it. Also, don’t use the inscribed-angle rule for a central angle.

🔄 Real-world variant

If the problem instead asked for the major arc intercepted by $\angle ABC$, you would compute $360^\circ-130^\circ=230^\circ$. If the angle at $C$ were an inscribed angle intercepting arc $AC$, then the arc would be $2\times130^\circ=260^\circ$ instead of $130^\circ$.

🔍 Related terms

arc measure, central angle, inscribed angle