Bellwork One-half of the measure of an angle plus its supplement is

equal to the measure of the angle. Find the measure of the angle

Solve for x 2x2-13x-7=0

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Bellwork Solution One-half of the measure of an angle plus its supplement is

equal to the measure of the angle. Find the measure of the angle

Solve for x 2x2-13x-7=0

APPLY OTHER ANGLES RELATIONSHIPS IN CIRCLES

Section 10.5

The Concept For the bulk of this chapter we’ve been dealing with the arc of

circles Today we’re going to finish out our discussion of arcs and

angles

TheoremTheorem 10.11

If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one half the measure of its intercepted arc

A

C

BD

This makes sense if we think about this in terms of

an inscribed angle with only

one leg

On your own What is the measure of arc ADC

82oA

C

B

D

.41

.82

.164

A

B

C

On your own What is the measure of arc ABC

82oA

C

B

D

.16

.164

.196

A

B

C

On your own What is the measure of angle ACB, if arc CD measures 160o

35o

A

C

B

D

.35

.65

.130

A

B

C

TheoremsTheorem 10.12

If two chords intersect inside a circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle

A

B

C

D

12m x mAB mCD

x

ExampleFind the measure of angle AEC

A

B

C

D

12m AEC mAC mBD

E

110o

96o

12 110 96

12 206

103

On your own What is the measure of angle ABC

225o

A

C

B

D

.30

.240

.480

A

B

C

E

255o

On your own What is the measure of angle ABD

148o

A

C

B

D

.89

.91

.178

A

B

C

E

30o

On your own What is the measure of arc AC

121o

A

C

B

D

.38

.81.5

.163

A

B

C

E

142o

TheoremsTheorem 10.13 Angles Outside the Circle Theorem

If a tangent and a secant, two tangents or two secants intersect outside a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs

A

B

C

D

12m x mCD mAB

TheoremsFind the measure of angle ADB

A

B

D

12m ADB mACB mAB

250o

110o

C 1

2 250 110

12 140

70

On your own What is the measure of angle ABD

30o

A

C

B

D

.28

.56

.58

A

B

C

E

86o

On your own What is the measure of angle ADB

65o

A

B

D .65

.130

.260

A

B

C

195o

On your own What is the measure of arc BD

32o

A

B

D

.98

.100

.132

A

B

C

164o

On your own What is the measure of arc AB

22o

A

B

D

.49

.98

.142

A

B

C

120o

On your own What is the measure of angle of E, if arc ABC is 200o

A

B

D

.20

.40

.100

A

B

C

C

E

On your own What is the measure of angle of F?A

B

D

.15

.30

.95

A

B

C

C

F

110o

80o

On your own What is the measure of angle of F?A

B

D

.35

.32.5

.45

.50

A

B

C

D

C

F

35o

110o

105o

Homework

10.51-20, 22-25

Most Important Points Angle-Chord Relationships that

intercept inside a circle Angle relationship that intersects

outside the circle