Question

which statements are true regarding the relationships between central, inscribed, and circumscribed angles of a circle? check all that apply.
a circumscribed angle is created by two intersecting tangent segments.
a central angle is created by two intersecting chords that are not a diameter.
the measure of a central angle will be twice the measure of an inscribed angle that intercepts the same arc.
the measure of a central angle will be half the measure of an inscribed angle that intercepts the same arc.
the measures of a central angle and circumscribed angle that intercept the same arc will sum to 90°.
the measure of a central angle will be equal to the measure of an inscribed angle when the arc intercepted by the inscribed angle is twice as large as the arc intercepted by the central angle.

Explanation:

1. For "A circumscribed angle is created by two intersecting tangent segments": A circumscribed angle (also called a tangent - tangent angle) is formed when two tangent segments intersect outside a circle. So this statement is true.
2. For "A central angle is created by two intersecting chords that are not a diameter": A central angle is formed by two radii (a radius is a chord that goes from the center to the circumference) with the vertex at the center of the circle. If the two chords are a diameter, the central angle is \(180^{\circ}\), but a central angle can also be formed by two non - diameter chords (radii) with the vertex at the center. Wait, actually, a central angle is defined as an angle whose vertex is the center of the circle and whose sides are radii of the circle. So two intersecting chords (radii) at the center form a central angle, regardless of whether they are a diameter or not. But the statement says "two intersecting chords that are not a diameter" - a central angle can be formed by a diameter (which is a chord) as well. However, the main idea is that a central angle has its vertex at the center and sides as radii (chords from center to circumference). But this statement is incorrect because a central angle can be formed by a diameter (a chord that passes through the center).
3. For "The measure of a central angle will be twice the measure of an inscribed angle that intercepts the same arc": The Inscribed Angle Theorem states that the measure of an inscribed angle is half the measure of the central angle that intercepts the same arc. So the central angle is twice the inscribed angle for the same arc. This statement is true.
4. For "The measure of a central angle will be half the measure of an inscribed angle that intercepts the same arc": This is the opposite of the Inscribed Angle Theorem. The inscribed angle is half the central angle, so this statement is false.
5. For "The measures of a central angle and circumscribed angle that intercept the same arc will sum to \(90^{\circ}\)": Let the central angle be \(C\) and the circumscribed angle be \(T\) that intercept the same arc. If the central angle intercepts arc \(AB\), and the circumscribed angle is formed by tangents from an external point \(P\) to \(A\) and \(B\), then the measure of the circumscribed angle \(T=\frac{1}{2}(\text{measure of the major arc}-\text{measure of the minor arc})\). If the central angle \(C\) intercepts the minor arc, then \(T = 180^{\circ}-\frac{C}{2}\) (derived from the property of tangents and central angles). So their sum is not \(90^{\circ}\) in general. This statement is false.
6. For "The measure of a central angle will be equal to the measure of an inscribed angle when the arc intercepted by the inscribed angle is twice as large as the arc intercepted by the central angle": Let the measure of the central angle be \(C\) (intercepting arc \(A\)) and the inscribed angle be \(I\) (intercepting arc \(B\)). We know that \(I=\frac{1}{2}\times\) measure of its intercepted arc and \(C=\) measure of its intercepted arc. If \(B = 2A\), then \(I=\frac{1}{2}\times B=\frac{1}{2}\times2A=A = C\). So this statement is true.

Answer:

A. A circumscribed angle is created by two intersecting tangent segments.
C. The measure of a central angle will be twice the measure of an inscribed angle that intercepts the same arc.
F. The measure of a central angle will be equal to the measure of an inscribed angle when the arc intercepted by the inscribed angle is twice as large as the arc intercepted by the central angle.