A radius drawn to a tangent at the point of tangency is perpendicular to the tangent. l C T Line l is tangent to Circle C at point T. CT  l at T

Two tangent segments drawn to a circle from the same exterior point are congruent. A C B O CA and CB are tangent segments. CA  CB

O A B CENTRAL ANGLE --VERTEX IS AT THE CENTER --SIDES ARE RADII --ITS MEASURE IS EQUAL TO THE MEASURE OF ITS INTERCEPTED ARC 1

O A B 1 The arc of an angle is the portion of the circle in the INTERIOR of the angle.

O A B C INSCRIBED ANGLE --VERTEX IS ON THE CIRCLE --SIDES ARE CHORDS --ITS MEASURE IS EQUAL TO HALF THE MEASURE OF ITS INTERCEPTED ARC

ANGLE FORMED BY CHORD(SECANT) AND TANGENT WITH VERTEX ON CIRCLE --ITS MEASURE IS EQUAL TO HALF THE MEASURE OF ITS INTERCEPTED ARC C 3 O A B

ANGLE FORMED BY TWO CHORDS (SECANTS) WHOSE VERTEX IS IN THE INTERIOR OF THE CIRCLE, BUT NOT AT THE CENTER --ITS MEASURE IS EQUAL TO HALF THE SUM OF THE MEASURES OF ITS INTERCEPTED ARCS (ITS ARC AND THE ARC OF ITS VERTICAL ANGLE) C A B D E

AN ANGLE WHOSE VERTEX IS IN THE EXTERIOR OF A CIRCLE MAY BE FORMED BY: A TANGENT AND A SECANT TWO SECANTS OR TWO TANGENTS B A C D B A C D A B C D E

IN EACH CASE, THE MEASURE OF THE ANGLE IS ONE-HALF THE DIFFERENCE OF THE MEASURES OF THE INTERCEPTED ARCS.

B A C D A B C D B A C D E

Two points on a circle will always determine:

A CHORD

TWO RADII

A MINOR ARC

A MAJOR ARC

OR TWO SEMICIRCLES

CONGRUENT CENTRAL ANGLES WILL HAVE CONGRUENT CHORDS AND ARCS (AND VICE-VERSA) IF AND ONLY IF THEY ARE IN THE SAME OR IN CONGRUENT CIRCLES!!

IN THE SAME CIRCLE OR IN CONGRUENT CIRCLES, CONGRUENT CHORDS ARE EQUIDISTANT FROM THE CENTER.

IN THE SAME CIRCLE OR IN CONGRUENT CIRCLES, IF TWO CHORDS ARE EQUIDISTANT FROM THE CENTER, THEN THEY ARE CONGRUENT.

IN THE SAME CIRCLE OR IN CONGRUENT CIRCLES, IF TWO CHORDS ARE UNEQUAL, THEN THE LONGER CHORD IS CLOSER TO THE CENTER.

IN THE SAME CIRCLE OR IN CONGRUENT CIRCLES, IF THE DISTANCES FROM THE CENTER OF TWO CHORDS ARE UNEQUAL, THEN THE LONGER CHORD IS CLOSER TO THE CENTER.

O “Anything” from the center of a circle (segment,radius,diameter) A B

O perpendicular to a chord A B

O A B bisects “everything” it touches:

O A B the chord C

O A B the central angle C

O A B the minor arc C

O A B the major arc C

Describe how the center,O, can be located. B A C D

Construct the perpendicular bisectors of the chords. They will intersect at the center! B A C D O

A B 1 2 IF TWO INSCRIBED ANGLES INTERCEPT THE SAME ARC, THEN THEY ARE CONGRUENT.

B O A C D IF AN ANGLE IS INSCRIBED IN A SEMICIRCLE, THEN IT IS A RIGHT ANGLE.

A B C D O IF A QUADRILATERAL IS INSCRIBED IN A CIRCLE, THEN ANY PAIR OF OPPOSITE ANGLES ARE SUPPLEMENTARY.

a b c d a b c b a c d

IF TWO CHORDS OF A CIRCLE INTERSECT, THEN THE PRODUCT OF THE LENGTHS OF THE SEGMENTS ON ONE CHORD EQUALS THE PRODUCT OF THE LENGTHS OF THE SEGMENTS ON THE OTHER CHORD. a b c d ab = cd

IF A TANGENT SEGMENT AND A SECANT SEGMENT INTERSECT IN THE EXTERIOR OF A CIRCLE, THEN THE SQUARE OF THE LENGTH OF THE TANGENT SEGMENT IS EQUAL TO THE PRODUCT OF THE LENGTHS OF THE SECANT SEGMENT AND ITS EXTERNAL PART. a b c a 2 = bc

IF TWO SECANT SEGMENTS INTERSECT IN THE EXTERIOR OF A CIRCLE, THEN THE PRODUCT OF THE LENGTHS OF ONE SECANT SEGMENT AND ITS EXTERNAL PART IS EQUAL TO THE PRODUCT OF THE LENGTHS OF THE OTHER SECANT SEGMENT AND ITS EXTERNAL PART. a b d c ab = cd