Download Theorems Of Circles

Chapter 10
Mr. Mills
Sum of Central Angles
 The 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 Arcs
 In 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 arcs
 Draw 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 Arcs
 In 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 Arcs
 Draw 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 Radius
 Draw 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 BM
 If AB has length 10 and the radius is 6 find EM, the
distance form the center.
You do the Math
M
X
RM = 8
XP = 3
Find MP
R
P
You do the Math
M
X
RX = 12
XP = 5
Find MP
Find XM
Find RM
R
P
You do the Math
M
X
If RM is
congruent to
ST , XP is 8,
and XM is 6.
R
P
Find PS
Find ST
Find Pl
S
L
T
Page 543
#7
Tell why the measure of angle CAM is 28 degrees.
Hint: Think SSS.
Page 543 # 8
 Explain 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 Angles
 An 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 Theorem
 If 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