The problem involves finding the angle $x$ in a circle with its center at $O$, where the angle at the center $O$ is given as $144^\circ$.

Step-by-Step Solution:

1. Understanding Circle Geometry: The angle $x$ lies on the circumference of the circle, and it subtends the same arc as the central angle $144^\circ$. By the Angle at the Center Theorem, we know that: $\text{The angle at the center is twice the angle at the circumference subtending the same arc.}$

2. Applying the Theorem: According to the theorem: $\angle O = 2 \times \angle x$ Substituting the given value: $144^\circ = 2 \times x$

3. Solving for $x$: $x = \frac{144^\circ}{2} = 72^\circ$

Conclusion

The size of angle $x$ is $72^\circ$.

Justification

This answer follows directly from the rule that the angle at the circumference is half of the angle at the center when both angles subtend the same arc.

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Related Questions:

1. What is the Angle at the Center Theorem, and how does it apply to circles?
2. How would you solve for an angle if the given central angle was different?
3. Can you explain how inscribed angles relate to the diameter of a circle?
4. What other properties of circles are useful in solving geometric problems?
5. How can we determine the measure of an exterior angle in a circle?

Tip:

Always remember that the angle subtended by the same arc at the circumference is half the angle subtended at the center of the circle.