Ohio Assessments for Educators (OAE) Mathematics Practice Exam

How is the measure of the angle defined when two secants intersect outside a circle?

1. Equal to the sum of the intercepted arcs

2. Equal to the difference of the two arcs that lie between the secants

3. Equal to the half of the difference of the two arcs

4. Equal to the half of the sum of the intercepted arcs

The correct answer is: Equal to the half of the difference of the two arcs

When two secants intersect outside a circle, the measure of the angle formed is equal to half of the difference of the measures of the intercepted arcs. This relationship can be established through an application of the properties of angles formed by intersecting chords, secants, and tangents in circles. The angle formed (let's denote it as Angle A) is calculated using the formula: \[ \text{Angle A} = \frac{1}{2} \times \left( \text{Arc 1} - \text{Arc 2} \right) \] where Arc 1 and Arc 2 are the measures of the arcs intercepted by the two secants. The reason this formula is relevant in this scenario is that the angle outside the circle is influenced by the difference between the two arcs that the secants intercept, resulting in an angle whose measure reflects this metric. This distinction is crucial because it shows how the angle measures relate directly to the arcs and differentiates it from other relationships such as those involving angles formed by chords or those formed at points on the circle. Understanding this concept is essential for solving problems related to angles outside circles and can aid in further geometry studies.