Circle Geometry: Inscribed Angles and Intersecting Chords

Vocabulary practice flashcards covering circle geometry theorems, arc-to-circumference proportionalities, inscribed angles, and intersecting chord relationships from textbook problems 8 and 9.

Inscribed Angle and Arc Ratio

The principle stating that the measure of an inscribed angle is directly proportional to the length of its subtended arc, where the inscribed angle for an arc comprising the full circumference equals $180^{\circ}$.

Intersecting Chords Angle Theorem

The theorem stating that the measure of an angle formed by two chords intersecting inside a circle equals the sum of the inscribed angles intercepting the two opposite arcs: $\angle x = \angle ADB + \angle CAD$.

Problem 8: Inscribed Angle for Arc AB

The measure of the inscribed angle subtending arc $AB$, calculated from the arc being $\frac{1}{5}$ of the circle's circumference: $\frac{1}{5} \times 180^{\circ} = 36^{\circ}$.

Problem 8: Inscribed Angle for Arc CD

The measure of the inscribed angle subtending arc $CD$, calculated using the ratio $\text{arc } AB : \text{arc } CD = 3 : 4$, which gives an arc fraction of $\frac{4}{15}$ and an inscribed angle of $\frac{4}{15} \times 180^{\circ} = 48^{\circ}$.

Problem 8: Measure of Angle x

The interior angle $\angle x$ formed at intersection point $P$ between chords $AC$ and $BD$, equal to the sum of the inscribed angles intercepting arcs $AB$ and $CD$: $36^{\circ} + 48^{\circ} = 84^{\circ}$.

Tangent-Chord Theorem (Problem 9)

The geometric theorem stating that the angle between a tangent line and a chord passing through the point of tangency is equal to the inscribed angle subtending that chord in the alternate segment.