Ohio Assessments for Educators (OAE) Mathematics Practice Exam

What is the measure of the vertical angles formed by two secants intersecting inside a circle?

1. Equal to the sum of the two intercepted arcs

2. Equal to half the sum of the two intercepted arcs

3. Equal to the difference of the two intercepted arcs

4. Equal to half the difference of the two intercepted arcs

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

The measure of the vertical angles formed by two secants intersecting inside a circle is indeed equal to half the sum of the two intercepted arcs. When two secants intersect inside a circle, they create two pairs of vertical angles. To understand why this is true, consider the concept of intercepted arcs defined by the two secants. Each secant divides the circle into arcs, and the vertical angles can be related directly to these arcs. The angle formed between the intersecting secants can be computed by looking at the arcs that are created by where these secants intersect the circle. The rule states that the angle formed by two intersecting secants inside the circle is given by taking the average of the measures of the arcs intercepted by these angles. This relationship holds true because vertical angles are congruent, and thus the angles formed respect the properties of circle geometry—they are linked directly to the arcs they intercept. Consequently, the correct answer reflects this relationship by asserting that the measure of the angle is half the sum of the measures of the intercepted arcs, effectively summarizing a fundamental principle of circle geometry.