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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.

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Theorems of circles

Theorems Of Circles

Chapter 10

Mr. Mills


Sum of central angles
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
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 arcs1
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
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 arcs1
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
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
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
You do the Math

M

X

RM = 8

XP = 3

Find MP

R

P


You do the math1
You do the Math

M

X

RX = 12

XP = 5

Find MP

Find XM

Find RM

R

P


You do the math2
You do the Math

M

X

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

Find PS

Find ST

Find Pl

R

P

L

S

T


Page 543
Page 543

#7

Tell why the measure of angle CAM is 28 degrees.

Hint: Think SSS.


Page 543 8
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 5431
Page 543

  • # 9

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


Inscribed angles
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 theorem
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


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