CIRCLES 2 Moody Mathematics. ANGLE PROPERTIES: Moody Mathematics Let’s review the methods for finding the arcs and the different kinds of angles found

CIRCLES 2CIRCLES 2

Moody MathematicsMoody Mathematics

ANGLE ANGLE PROPERTIES:PROPERTIES:


Let’s review the Let’s review the methods for finding methods for finding the arcs and the the arcs and the different kinds of different kinds of angles found in angles found in circles.circles.


The measure of a minor arc is the same as…

…the measure of its central angle.


75 75

Example:


The measure of an inscribed angle is…

…half the measure of its intercepted angle.


88

44

Example:


The measure of an angle formed by a tangent and secant is …

…half the measure of its intercepted arc.


Example:

230115

130

65


The measure of one of the vertical angles formed by 2 intersecting chords

...is half the sum of the two intercepted arcs.


Example:110

60

85

1(110 60 )2


The measure of an angle formed by 2 secants intersecting outside of a circle is…

…half the difference of the measures of its two intercepted arcs.


Example:

90

20

35 1(90 20 )2


The measure of an angle formed by 2 tangents intersecting outside of a circle is…

…half the difference of the measures of its two intercepted arcs.


Example:

250

110

70 1(250 110 )2

PROPERTIES: PROPERTIES: Complete the Complete the theorem relating theorem relating the objects the objects pictured in each pictured in each frame.frame.


Note: Note: Many Many of our theorems of our theorems begin the same begin the same way, “In the same way, “In the same circle, circle, or in or in congruent congruent circlescircles…”…”


So: So: We will We will just start “In the just start “In the same circle*…” same circle*…” where the where the ** represents the represents the rest of the phrase. rest of the phrase.



All radii in the same circle,* …

...are congruent.


In the same circle,* Congruent central angles...

...intercept congruent arcs.


In the same circle,* Congruent Chords...

...intercept congruent arcs.


Tangent segments from an exterior point to a circle…

...are congruent.


The radius drawn to a tangent at the point of tangency…

...is perpendicular to the tangent.


If a diameter (or radius) is perpendicular to a chord, then…

...it bisects the chord……and the arcs.


In the same circle,* Congruent Chords...

...are equidistant from the center.


Example: Given a circle of radius 5” and two 8” chords. Find their distance to the center.

5

4

4

2 2 24 5x 3x

x


If two Inscribed angles intercept the same arc...

...then they are congruent.


If an inscribed angle intercepts or is inscribed in a semicircle …

...then it is a right angle.

180


If a quadrilateral is inscribed in a circle then each pair of opposite angles …

...must be supplementary.

(total 180o)


If 2 chords intersect in a circle, the lengths of segments formed have the following relationship:

a b c d

a

b

c d


3 4 2 c 3

4

c

Example:

2

6c


If 2 secants intersect outside of a circle, their lengths are related by…

a b c d

a

c

bd


8 3 2c

c

32

Example:

8

12c


If a secant and tangent intersect outside of a circle, their lengths are related by…

a a c d

a

c

d


4 (4 5)a a a

5

4

Example:

6a

Let’s Let’s Practice!Practice!


Example: Given

50

P

A

B

C

D

P

mAB

mBC

mABC

mADB

mACD

50

130

180

310

230


Example:

200100

160

80

x

y

z


55 55

Example:


Example:

80

30

25 1(80 30 )2


Example: Given a circle of radius 13” and two 24” chords. Find their distance to the center.

1312

2 2 212 13x 5x

x12

Moody MathematicsMoody Mathematics110

55

35

70

x180

y

z

Example:


5 (15 5)x x x

15

5

Example:

10x


Example:110

40

75

1(110 40 )2

x


Example:

230

130

50 1(230 130 )2


140

x

160

y40

120

Example:


8 12 6x

x

12 68

9x

Example:

Example: Of the following quadrilaterals, which can not always be inscribed in a circle?

A.Rectangle

B.Rhombus

C.Square

D.Isosceles Trapezoid


90

50

x

y

z

25x

45y

70z

Example:


Example: 160

xy

z

80x

100y

50z


Example: Regular Hexagon ABCDEF is inscribed in a circle. A B

C

DE

F

mACE 240

THE END!THE END!Now go practice!

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CIRCLES 2 Moody Mathematics. ANGLE PROPERTIES: Moody Mathematics Let’s review the methods for finding the arcs and the different kinds of angles found