Chapter 10 - Geometry

the set of all points in a plane that are equidistant from a given point, center of a circle.

in a plane, a line is tangent to a circle if and only if the line is perpendicular to a radius of the circle at its endpoint on the circle

tangent segments from a common external point are congruent

a segment whose endpoints lie ON a circle

line that intersects a circle at two points

line in the same place as a circle that intersects it at exactly 1 point

point where the tangent and a circle intersect

congruent radii

coplaner circles with common center

coplaner circles that intersect at 1 point only

line tangent two two circles

an angle whose vertex is the center of the circle

an unbroken piece of a circle consisting of two points (endpoints) and all points on the circle between them

< 180

180

= 180

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

two circles are congruent circles if and only if they have the same radius

in the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding central angles are congruent

all circles are similar

in the same circle or congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent

if a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc

if one chord of a circle is a perpendicular bisector of another chord, then the first chord is the diameter (defines diameter!)

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

RADIUS’ AND TRIANGLES!!!

an angle whose vertex is on a circle and whose sides contain chords of the circle

an arc that lies between two lines, rays, or segments

circle that contains a polygon is an inscribed polygon when all its vertices lie on a circle. the circle that contains the vertices is a circumscribed 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

if a right triangle is inscribed in a circle, then the hypotenuse is the diameter of the circle. Conversely, if one side of an inscribed triangle is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is the right angle

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

if a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is 1/2 the measure of its intercepted arc

if two chords intersect INSIDE a circle, then the measure of each angle is 1/2 the SUM of the measure of the arcs intercepted by the angle and its vertical angle (opposite arcs)

if a tangent and a secant, two tangents, or two secants intersect OUTSIDE a circle, then the measure of the angle formed is 1.2 the DIFFERENCE of the measures of the intercepted arcs (LOOK @ NOTES!!!!!!!!)

an angle whose sides are tangent to a circle

the measure of a circumscribed angle is equal to 180 degrees minus the measure of the CENTRAL angle that intercepts the same arc

segment that is tangent to a circle at an endpoint

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

part outside of the circle

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 (piece x piece = piece x piece)

if two secant segments share the same endpoint outside a circle, then the product of the lengths of one secant segment and its external segment equals the product of the lengths of the other secant segment and its external segment (outside piece x whole = outside piece x whole)

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 (tangent^2 = outside piece x whole)