How is the angle outside a circle calculated?

To find the angle outside a circle, you use the half the difference of the measures of the larger arc and the smaller arc that are intercepted by the angle. This relationship is based on the properties of angles formed by secants and tangents that intersect outside the circle.

When two lines (like secants) intersect outside the circle, the angle formed is indeed calculated by taking half the difference between the measures of the arcs they extend to. Specifically, if one arc measures greater than another, the formula used will yield the correct angle measurement outside the circle.

For instance, if the larger arc measures 120 degrees and the smaller arc measures 40 degrees, the angle outside the circle can be calculated as follows:

Angle = 1/2 (larger arc - smaller arc) = 1/2 (120 - 40) = 1/2 (80) = 40 degrees.

This demonstrates why the calculation of the angle outside a circle relies specifically on the difference between the arc measures. Other options provided, like half the sum, twice the smaller arc, or using only the larger arc, do not adhere to the established geometric principles regarding angles formed by lines intersecting outside a circle.