Geometry Basics: Circles and Angles

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Questions and Answers

What is an inscribed angle?

An angle whose vertex is inside the circle

An angle whose vertex is on a circle and sides contain chords (correct)

An angle whose vertex is at the center of the circle

An angle whose vertex is outside the circle

What is an intercepted arc?

An arc that lies between two lines, rays, or segments.

What does it mean to subtend an angle?

A segment or arc subtends an angle if its endpoints lie on the sides of the angle.

What is the Measure of an Inscribed Angle Theorem?

The measure of an inscribed angle is one half the measure of its intercepted arc. Signup and view all the answers

What does the inscribed angles of a circle theorem state?

If two inscribed angles of a circle intercept the same arc, then the angles are congruent. Signup and view all the answers

What is an inscribed polygon?

A polygon whose vertices all lie on a circle. Signup and view all the answers

What is a circumscribed circle?

The circle that contains the vertices of an inscribed polygon. Signup and view all the answers

What does the Inscribed Right Triangle Theorem state?

If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle. Signup and view all the answers

What does the Inscribed Quadrilateral Theorem state?

A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary. Signup and view all the answers

What is the tangent and intersected chord theorem?

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. Signup and view all the answers

How do intersecting lines and circles interact?

If two nonparallel lines intersect a circle, there are three places of intersection: on the circle, inside the circle, and outside the circle. Signup and view all the answers

What does the Angles Inside the Circle Theorem state?

If two chords intersect inside a circle, then each angle's measure is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle. Signup and view all the answers

What is the angles outside the circle theorem?

If a tangent and a secant, two tangents, or two secants intersect outside a circle, the angle formed is one half the difference of the measures of the intercepted arcs. Signup and view all the answers

What is a circumscribed angle?

An angle whose sides are tangent to a circle. Signup and view all the answers

What does the circumscribed angle theorem state?

The measure of a circumscribed angle is equal to 180° minus the measure of the central angle that intercepts the same arc. Signup and view all the answers

What does the segments of a chord theorem state?

If two chords in a circle intersect, 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 second chord. Signup and view all the answers

What is a tangent segment?

A segment of a tangent with one endpoint on the circle. Signup and view all the answers

What is a secant segment?

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

What is the external segment of a secant?

The part of a secant segment that is outside the circle. Signup and view all the answers

What does the segments of secants theorem state?

If two secant segments share the same endpoint outside a circle, the product of one secant segment and its external segment equals the product of the other secant segment and its external segment. Signup and view all the answers

What does the Segments of Secants and Tangents Theorem state?

If a secant segment and a tangent segment share an endpoint outside a circle, then the product of the lengths of the secant segment and its external segment equals the square of the length of the tangent segment. Signup and view all the answers

What is the standard equation of a circle?

(x-h)^2 + (y-k)^2 = r^2. Signup and view all the answers

What defines a circle?

The set of all points in a plane that are the same distance from a given point called the center. Signup and view all the answers

What is the center of a circle?

Point inside a circle that is the same distance from every point on the circle. Signup and view all the answers

What is a chord?

A segment whose endpoints lie on a circle. Signup and view all the answers

What is a radius?

The distance from the center of a circle to any point on the circle. Signup and view all the answers

What is a diameter?

A chord that passes through the center of the circle. Signup and view all the answers

What is a secant?

A line that intersects a circle in two points. Signup and view all the answers

What is a tangent?

A line in the plane of a circle that intersects the circle in exactly one point. Signup and view all the answers

What is a point of tangency?

The point where a circle and a tangent intersect. Signup and view all the answers

What are tangent circles?

Two coplanar circles that intersect at exactly one point. Signup and view all the answers

What are concentric circles?

Circles that lie in the same plane and have the same center. Signup and view all the answers

What is a common tangent?

A line or segment that is tangent to two coplanar circles. Signup and view all the answers

What does the Tangent Line to Circle Theorem state?

A line is tangent to a circle if and only if it is perpendicular to a radius of the circle at its endpoint on the circle. Signup and view all the answers

What does the External Tangent Congruence Theorem state?

Tangent segments from a common external point are congruent. Signup and view all the answers

What is a central angle?

An angle whose vertex is the center of the circle. Signup and view all the answers

What is a minor arc?

An arc of a circle whose measure is less than 180 degrees. Signup and view all the answers

What is a major arc?

An arc of a circle whose measure is greater than 180 degrees. Signup and view all the answers

What is a semicircle?

An arc of a circle whose endpoints lie on a diameter. Signup and view all the answers

What is the measure of a minor arc?

The measure of its central angle. Signup and view all the answers

What is the measure of a major arc?

The difference between 360 and the measure of the related minor arc. Signup and view all the answers

What are adjacent arcs?

Arcs of the same circle that have exactly one point in common. Signup and view all the answers

What does the Arc Addition Postulate state?

The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. Signup and view all the answers

What does the Congruent Circles Theorem state?

Two circles are congruent circles if and only if they have the same radius. Signup and view all the answers

What are congruent circles?

Circles are congruent if and only if a rigid motion or composition of rigid motion maps one circle onto the other. Signup and view all the answers

What does the Congruent Central Angles Theorem state?

In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding central angles are congruent. Signup and view all the answers

What does the Similar Circles Theorem state?

All circles are similar. Signup and view all the answers

What does the Congruent Corresponding Chords Theorem state?

In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent. Signup and view all the answers

What does the Perpendicular Chord Bisector Theorem state?

If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc. Signup and view all the answers

What does the Perpendicular Chord Bisector Theorem converse state?

If one chord of a circle is a perpendicular bisector of another chord, then the first chord is a diameter. Signup and view all the answers

What does the Equidistant Chords Theorem state?

In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center. Signup and view all the answers

Study Notes

Geometry Basics

A circle is the set of all points in a plane that are the same distance from a given point called the center.
The center of a circle is the point inside a circle, equidistant from every point on the circle.

Angles and Arcs

An inscribed angle is an angle whose vertex is on a circle and whose sides contain chords of the circle.
The measure of an inscribed angle is one half the measure of its intercepted arc.
If two inscribed angles of a circle intercept the same arc, then the angles are congruent.
An intercepted arc is an arc that lies between two lines, rays, or segments.
A segment or arc subtends an angle if the endpoints of the segment or arc lie on the sides of the angle.

Inscribed Polygons and Circles

An inscribed polygon is a polygon whose vertices all lie on a circle.
A circumscribed circle is the circle that contains the vertices of an inscribed polygon.
A circle can be inscribed in a quadrilateral if and only if its opposite angles are supplementary.
If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle.

Tangents and Secants

A tangent is a line in the plane of a circle that measures the circle in exactly one point.
A secant is a line that intersects a circle in two points.
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.
If two nonparallel lines intersect a circle, there are three places where the lines can intersect: on the circle, inside the circle, outside the circle.

Circles and Intersecting Lines

If two chords intersect inside a circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle.
If a tangent and a secant, two tangents, or two secants intersect outside a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs.
The measure of a circumscribed angle is equal to 180° minus the measure of the central angle that intercepts the same arc.

Segments of Chords and Circles

When two chords intersect in the interior of a circle, each chord is divided into two segments.
If two chords in a circle intersect, 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 second chord.
If a secant segment and a tangent segment share an endpoint outside a circle, then the product of the lengths of the secant segment and its external segment equals the square of the length of the tangent segment.

Standard Equation of a Circle

The standard equation of a circle is (x-h)^2 + (y-k)^2 = r^2.

Circle Properties

A radius is the distance from the center of a circle to any point on the circle.
A diameter is a chord that passes through the center of the circle.
A central angle is an angle whose vertex is the center of the circle.
A minor arc is an arc of a circle whose measure is less than 180 degrees.
A major arc is an arc of a circle whose measure is greater than 180 degrees.
A semicircle is an arc of a circle whose endpoints lie on a diameter.

Arcs and Chords

The measure of a minor arc is the measure of its central angle.
The measure of a major arc is the difference between 360 and the measure of the related minor arc.
Adjacent arcs are arcs of the same circle that have exactly one point in common.
The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.

Congruent Circles and Arcs

Two circles are congruent if and only if they have the same radius.
Congruent circles are circles that are congruent if and only if a rigid motion or composition of rigid motions maps one circle onto the other.
In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding central angles are congruent.
In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.

Similar Circles and Chords

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

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Description

Learn about the fundamentals of circles, angles, and arcs, including inscribed angles, intercepted arcs, and inscribed polygons. Understand the relationships between circles and polygons.