Circle Theorems and Vocabulary

Vocabulary flashcards covering key definitions and theorems about circles, tangents, chords, and angle relations from the lecture notes.

Coplanar Circles

Two or more circles that lie in the same plane.

Tangent Circles

Coplanar circles that intersect in exactly one point.

Concentric Circles

Coplanar circles that share the same center but have different radii.

Common Tangent

A line, ray, or segment that is tangent to two coplanar circles.

Tangent–Radius Theorem

A line is tangent to a circle if and only if it is perpendicular to a radius drawn to the point of tangency.

Tangent Segments Theorem

Two tangent segments drawn from the same external point to a circle are congruent.

Congruent Chords and Arcs

In the same or congruent circles, minor arcs are congruent if and only if their corresponding chords are congruent.

Perpendicular Bisector–Diameter Theorem

If one chord is the perpendicular bisector of another chord, the first chord is a diameter of the circle.

Arc Addition Postulate

The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.

Diameter Perpendicular to a Chord

A diameter that is perpendicular to a chord bisects both the chord and its intercepted arc.

Equidistant Chords Theorem

In the same or congruent circles, two chords are congruent if and only if they are equidistant from the center.

Measure of an Inscribed Angle

The measure of an inscribed angle equals one-half the measure of its intercepted arc.

Inscribed Angles Intercepting Same Arc

If two inscribed angles intercept the same arc, the angles are congruent.

Right Triangle–Diameter Theorem

An inscribed right triangle has its hypotenuse as a diameter, and conversely, any triangle with a side as a diameter is a right triangle.

Inscribed Quadrilateral Criterion

A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary.

Tangent–Chord Angle Theorem

An angle formed by a tangent and a chord at the point of tangency measures one-half its intercepted arc.

Angles Inside the Circle Theorem

When two chords intersect inside a circle, each angle measures one-half the sum of the arcs intercepted by the angle and its vertical angle.

Angles Outside the Circle Theorem

An angle formed by a tangent and a secant, two tangents, or two secants outside a circle measures one-half the difference of the intercepted arcs.

External Segment

The part of a secant segment that lies outside the circle.

Secant Segment

A segment that contains a chord of the circle and has exactly one endpoint outside the circle.

Tangent Segment

A segment that is tangent to the circle at one endpoint and has its other endpoint outside the circle.

Segments of Chords Theorem

If two chords intersect inside a circle, the product of the lengths of the segments of one chord equals the product of the lengths of the segments of the other chord.

Segments of Secants Theorem

For two secants from the same external point, the product of a secant
amen's entire length and its external segment equals the corresponding product for the other secant.

Secant–Tangent Theorem

For a secant and a tangent from the same external point, the product of the secant
amen's length and its external segment equals the square of the tangent segment
amen's length.