Theorems Of CirclesChapter 10

Mr. Mills

Sum of Central AnglesThe sum of the measures of the central agles

of a circle with no interior points in common is 360 degrees.

Draw circle p with radii PA, PB, PC. The sum of the measures of angles APB, BPC, CPA is 360 degrees.

Congruent ArcsIn the same circle or congruent circles, two

arcs are congruent if and only if their corresponding central angles are congruent.

Draw Circle P with radii PA and PB. Draw chord AB.

Draw an angle CPD that is congruent to central angle APB.

Congruent arcsDraw circle E with congruent angles RED

and SET.

What do you know about Minor arcs RD and ST ?

If arc ST has a length of 27 inches, what is the length of arc RD?

Congruent Minor ArcsIn a circle or congruent circles, two minor

arcs are congruent, if and only if their corresponding chords are congruent.

Draw circle P with chord AB and Chord CD. So that the two chords are congruent.

Congruent Minor ArcsDraw circle E with congruent chords AB and

CD.What do we know about arcs AB and CD?If the measure of arc AB is 56 degrees, what

is the measure of arc CD?

Diameter or Radius In a circle, if a diameter or radius is

perpendicular to a chord, then it bisects the chord and its arc.

Draw circle P with chord AB and radius CP that is perpendicular to chord AB.

Diameter and RadiusDraw circle E with radius EZ perpendicular

to chord AB. Label the intersection of the chord and radius as point M.

IF AB has length 10,Find AM and BMIf AB has length 10 and the radius is 6 find

EM, the distance form the center.

You do the Math

P

X

M

R

RM = 8XP = 3

Find MP

You do the Math

P

X

M

R

RX = 12 XP = 5

Find MP

Find XM

Find RM

X

M

R

S T

You do the Math

P

If RM is congruent to ST , XP is 8, and XM is 6.

Find PS

Find ST

Find Pl

L

Page 543#7Tell why the measure of angle CAM is 28

degrees.

Hint: Think SSS.

Page 543 # 8Explain how to show that the measure of arc

ES is 100 degrees.

Hint: The sum of interior angles of a triangle is 180 degrees.

Page 543# 9

Explain how to show the length of SC is 21 units.

Inscribed AnglesAn inscribed angle is an angle that has its

vertex on the circle and its sides contained in chords of the circle.

Inscribed angles Intercepted Arc

Inscribed Angles TheoremIf an angle is an inscribed angle, then the

measure is equal to ½ the measure of the intercepted arc or(the measure of the intercepted arc is twice the measure of the inscribed angle.

Inscribed angles