Theorems of circles
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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

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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 intercepted arc

Inscribed angles Intercepted Arc


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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