Math Ch. 10 Theorems and Vocab

If two inscribed or tangent - chord angles intercept the same arc, then they are congruent.

Theorems?

If two inscribed or tangent-chord angles intercept congruent arcs, then they are congruent.

Theorems?

The sum of the measures of a tangent-tangent angle and its minor intercepted arc is 180 degrees. measure of angle M + measure of arc NP = 180. MNOP is a kite. Angles add to 360. Measure of angle M + measure of angle O = 180.

Theorems?

each side of the polygon is tangent to the circle.

a polygon is circumscribed about a circle if..

each side of the polygon is tangent to the circle

A circle is inscribed in a polygon if

all of the polygon's vertices are on the circle.

A cirlce is circumscribed about a polygon if...

all of its vertices are on the circle

A polygon is inscribed in a cirlce if...

arcs ≅ --> central angles ≅

Describe the relationship in each diagram

arcs ≅ --> chords ≅

Describe the relationship in each diagram

central angles ≅ --> arcs ≅

Describe the relationship in each diagram

central angles ≅ --> chords ≅

Describe the relationship in each diagram

chords ≅ --> arcs ≅

Describe the relationship in each diagram

chords ≅ --> central angles ≅

Describe the relationship in each diagram

If 2 points are equidistant from endpoints of a seg, then perpendicular bisector.

If 2 points are equidistant from endpoints of a seg, ----

If 2 tangent segs are drawn to a circle from the same exterior point, then they are congruent.

If 2 tangent segs are drawn to a circle from the same exterior point....

If a radius bisects a chord, that is not a diameter, then it is perpendiuclar to the chord

If a radius bisects a chord, that is not a diameter,

If a radius of a circle is perpendicular to a chord, then it bisects that chord

If a radius of a circle is perpendicular to a chord, ----

if central angles congruent <=> arcs congruent <=> chords congruent

If same circle or congruent angles..

sometimes

If two arcs have the same measure, then they are (A/S/N) congruent

If two chords of a circle are congruent, then they are equidistant form the center of the circle

If two chords of a circle are congruent....

If two chords of a circle are equidistant from the center, then they are congruent.

If two chords of a circle are equidistant from the center...

A central angle

Is an angle whose vertex is at the center of a circle

90

Measure of AB

120

Measure of AB?

the major arc

Measure of AXB

or circle R is inscribed in the pentagon. Moreover, R is the incenter of LMNOP

Pentagon LMNOP is circumscribed about circle R,

Circle O is circumscribed about the quad. Moreover, O is the circumcenter of ABCD

Quad ABCD is inscribed in circle O,

180

measure of semicircle

center of the polygon. The center is equidistant to the sides of the polygon.

The center of a circle is called the..

circumcenter of the polygon. A circumcenter is equidistant to the vertices of the polyogn.

The center of the circle is called the

The perpendicular bisector of a chord passes through the center of the circle

The perpendicular bisector of a chord .....

(tangent segment)^2 = external segment x secant segment a^2 = c (b+c)

Theorem for Tangent - secant

two tangents segments from the same external point are congruent.

Theorem for tangent - secant

If a parallelogram is inscribed in a circle, it must be a rectangle. reasoning: 1. oppostie angles of a parallelogram are congruent 2. opposite angles of an inscribed quad are supp.

Theorems?

If a quadrilateral is inscribed in a cirlce, its opposite nagles are supplementary. measure of angle BAD = 1/2 of arc BC Measure of angle BCD = 1/2 of arc BAD

Theorems?

If an angle is inscribed in a semicricle, then it is a right angle

Theorems?

- opposite angles of a quadrilateral inscribed in a circle are supplementary

angles and quadrilaterals

- angle inscribed in a semicircle is a right angle

angles and semicircles

- measure of arc is same as measure of central angle

arcs and angles

area sector - area of triangle

area of circle segment

(x / 360) x area

area of sector

x = (arc of measure AB)

central angle

x = 1/2 (arc of mAB + arc of mCD)

chord chord angle

chords equidistant from the center are congruent

chords and center

chord through point of tangency perpendicular to tangent passes through center

chords and tangents

x = arc measure. (x / 360) x circumference

length of arc

arc

consists of two points on a circle and all points on the circle needed to connect the points by a single path.

same measure congruent, or same circle

definition of congruent arcs

central angles ≅ <--> chords ≅ <--> arcs ≅

describe relationship between central angles, chords, and arcs

diameter is longest chord

diameter and chord

A = 𝛑r²

formula of area

C = 𝛑d

formula of circumference

if a line is perpendicular to a radius at its outer endpoint, then it is tangent to the circle

if a line is perpendicular to a radius at its outer endpoint....

Perpendicular line segment

line from the center of the circle to a chord

if line perpendicular to radius at outer endpoint => tangent

line perpendicular to radius

x = 1/2 (arc of measure AB)

inscribed angle

semicircle

is an arc whose endpoints are the endpoints of a diameter

Minor arc

is an arc whose points are on or between the sides of a central angle (less than 180)

Major arc

is an arc whose points are on or outside of a central angle (more than 180)

if a line from center is perpendicular to chord <=> it bisects the chord

perpendicular and chords

perpendicular bisector of a chord passes through center of the circle

perpendicular bisector and center

x = 1/2 (arc of mAB - arc of mCD)

secant secant angle

x = 1/2 (mAB - mAC)

secant tangent angle

if tangent => perpendicular to radius

tangent and radius

x = 1/2 (arc of measure AB)

tangent chord

x = 1/2 (arc of mACB - arc of mAB)

tangent tangent angle

tangent-tangent angle and its intercepted arc are supplementary

tangent tangent angles and intercepted arc

Product of segments of one chord = product of segments of other chord ad = bc

theorem for chord-chord

external segment x secant segment = external segment x secant segment b(a+b) = d(c+d)

theorem for secant - secant

measure of angle x. = 1/2 sum intercepted arcs measure of angle x = 1/2 (measure of arc AB + measure of arc CD)

vertex inside circle

measure of angle x = 1/2 measure intercepted arc measure of angle x = 1/2 measure of arc AB

vertex on circle

measure of angle x = 1/2 difference intercepted arcs measure of angle x = 1/2 (measure of arc AB - measure of arc CD)

vertex outside circle

a tangent line is perpendicular to a radius drawn to the point of tangency

when is a tangent line perpendicular to its radius?