Geometry of Circles: Arcs, Chords, Tangents, and Secants

Vocabulary terms and geometric theorems covering circle properties, arcs, chords, angles, tangents, and secants from the review lecture.

Perpendicular Tangent Theorem

A geometric theorem stating that if a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency ($l \perp \overline{QP}$).

Central Angle

An angle whose vertex is at the center of a circle.

Minor Arc

A part of a circle that measures less than $180^\circ$, typically identified and named using two points.

Major Arc

A part of a circle that measures between $180^\circ$ and $360^\circ$, typically identified and named using three points.

Semicircle

An arc whose endpoints are the endpoints of a diameter of the circle, measuring exactly $180^\circ$ and named using three points.

Arcs and Chords Theorem

A theorem stating that in the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent ($\widehat{AB} \cong \widehat{BC} \iff \overline{AB} \cong \overline{BC}$).

Perpendicular Diameter Theorem

A theorem stating that if a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its intercepted arc ($\overline{DE} \cong \overline{EF}$ and $\widehat{DG} \cong \widehat{FG}$).

Perpendicular Diameter Converse

A theorem stating that if one chord is a perpendicular bisector of another chord, then the first chord is a diameter of the circle.

Congruent Chords Theorem

A theorem stating that in the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center ($\overline{AB} \cong \overline{CD} \iff \overline{EF} \cong \overline{EG}$).

Congruent Tangent Segments Theorem

A theorem stating that if two segments from the same exterior point are tangent to a circle, then the segments are congruent ($\overline{SR} \cong \overline{ST}$).

Inscribed Angle

An angle whose vertex lies on a circle and whose sides contain chords of the circle.

Intercepted Arc

The arc that lies in the interior of an inscribed angle and has endpoints lying on the rays of the angle.

Measure of an Inscribed Angle Theorem

A theorem stating that if an angle is inscribed in a circle, then its measure is equal to one half the measure of its intercepted arc ($m\angle ADB = \frac{1}{2}m\widehat{AB}$).

Congruent Inscribed Angles Theorem

A theorem stating that if two inscribed angles of a circle intercept the same arc, then the angles are congruent ($\angle C \cong \angle D$).

Inscribed Right Triangle Theorem

A theorem stating that if a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle; conversely, if one side of an inscribed triangle is a diameter, then the triangle is a right triangle and the angle opposite the diameter is the right angle.

Inscribed Quadrilateral Theorem

A theorem stating that a quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary ($m\angle D + m\angle F = 180^\circ$ and $m\angle E + m\angle G = 180^\circ$).

Tangent-Chord Theorem

A theorem stating that if a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one half the measure of its intercepted arc ($m\angle 1 = \frac{1}{2}m\widehat{AB}$).

Interior Intersection Theorem

A theorem stating that if two chords intersect in the interior of a circle, then the measure of each angle formed is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle ($m\angle 1 = \frac{1}{2}(m\widehat{CD} + m\widehat{AB})$).

Exterior Intersection Theorem

A theorem stating that if a tangent and a secant, two tangents, or two secants intersect in the exterior of a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs.

Chord Product Theorem

A theorem stating that if two chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord ($EA \cdot EB = EC \cdot ED$).

Secant-Secant Theorem

A theorem stating that if two secant segments share the same endpoint outside a circle, then the product of the length of one secant segment and its external segment equals the product of the length of the other secant segment and its external segment ($EA \cdot EB = EC \cdot ED$).

Secant-Tangent Theorem

A theorem stating that if a secant segment and a tangent segment share an endpoint outside a circle, then the product of the length of the secant segment and its external segment equals the square of the length of the tangent segment ($(EA)^2 = EC \cdot ED$).