Warm Up Section 4.5 Find x:

1. 2. 3.

4. 5. 6.

xo

xo

70o

32o

xo

xo100ox 12

xo

45o

Answers to Warm Up Section 4.5 Find x:

1. 2. 3.

4. 5. 6.

xo

xo

70o

32o

xo

xo100ox 12

xo

45o

90o 140o 32o

40o 45o 12 2

Angles Formed byChords, Secants, and Tangents

Section 4.5

Standard: MM2G3 bd

Essential Question: How are properties of chords, tangents, and secants used to find angle measures?

Recall how arcs are related to central angles:

arc = angle

Ex.

oxox

0x065

x = 65

Recall how an arc is related to its inscribed angle:

o40

ox

angle = ½ arc

Ex . Ex .

ox

ox2

ox

o64

arc = angle × 2angle = arc ÷ 2

x = 80 x = 32

The measure of the angle formed by two chords that intersect inside a circle is equal to half the sum of the measures of the intercepted arcs.

1X

Y

ZW

angle = ½ (arc 1 + arc 2)

11

2m mWY mXZ

Example 1:

If = 45o and = 75o, find m 1

S

1

R

P

Q

mRSmPQ

m1 = ½ (mRS + mPQ)m1 = ½(75o + 45o)m1 = ½(120o)m1 = 60o

75o45o

If and 80o , find

m1 = ½ (mRS + mPQ) 55o = ½(80o + x) 110o = (80o + x) 30o = x

Example 2:

S

R

P

Q

1 55m mRS .mPQ

80o55ox

x°

80°

20°100°

x°

Try these with your partner:3. 4.

xo = ½(80o + 20o)xo = ½(100o)xo = 50o

90o = ½(100o + xo)180o = (100o + xo )80o = xo

70°

x°

40°

5.

yo = ½(70o + 40o)yo = ½(110o)yo = 55o

xo = 180o – 55o = 125o

yo

Case 2: Vertex outside the circle.The measure of an angle formed by 2 secants, 2 tangents, or a secant and a tangent is half the difference of the measures of the intercepted arcs.

angle = ½ (arc 1 – arc 2)

A

CB

D

E

Example 6: If mDC = 100o and mEB = 40o, find mA.

100o 40o

mA = ½ (mDC – mEB)xo = ½(100o – 40o)xo = ½ (60o)xo = 30o

xo

W

VX

Y

Z

Example 7: If mW = 65o and mXZ = 70o, find mXVY

mW = ½ (mXVY – mXZ) 65o = ½(xo – 70o) 130o = xo – 70o

200o = xo

70o 65o

xo

P

Q

RS

Example 8: If mQRS = 240o, find mQS and mP.

mP = ½ (mQRS – mQS)xo = ½(240o – 120o)xo = ½ (120o)xo = 60o

240o

mQS = 360o – 240o

= 120o

120o xo

x°

230°

80°

140°x°

120°

Try these with your partner:9. 10.

xo = ½(80o – 20o)xo = ½ (60o)xo = 30o

xo = ½(230o – 130o)xo = ½ (100o)xo = 50o

130o

20o

x°

150°

35°

30°

x°

100°

11. 12.

xo = ½(50o – 30o)xo = ½ (20o)xo = 10o

35o = ½(150o – xo) 70o = (150o – xo) -80o = – xo

80o = xo

50o

1

x°

The vertex of the angle is located at the center of the circle. So, the angle is a central angle and is equal to the measure of the intercepted arc.

m1 = xo

Summary: Measures of Angles Formed by Radii, Chords, Tangents and Secants

angle = arc

2

x°

2

x°

The vertex of the angle is a point on the circle.So, the measure of the angle is one half the measure of the intercepted arc.

angle = ½arc m2 = ½ xo

3 x°y°

The vertex of the angle is located in the interior of the circle and not at the center, so the measure of the angle is half the sum of the intercepted arcs.

angle = ½(arc1 + arc2) m3 = ½(xo + yo)

4

x°

y°

4

x°

y°

4

x°

y°

The vertex of the angle is located in the exterior of the circle and not at the center, so the measure of the angle is half the difference of the intercepted arcs.

angle = ½(arc1 – arc2) m4 = ½(xo – yo)

3 O

B

4 1

2

9 6

78

5

10 A

C

D

E

BE is a diameter of the circle with center O. AT is tangent to the circle at A. mAB = 80o, mBC = 20o, and mDE = 50o.

80o

20o

50o

100o

110o

3 O

B

4 1

2

9 6

78

5

10 A

C

D

E

80o

20o

50o

100o

110o

13. m1 = ½(80o) = 40o

14. m2 = ½(100o) = 50o

15. m3 = ½(80o+ 50o) = 65o

16. m4 = ½(100o – 50o) = 25o

17. m5 = ½(80o) = 40o 18. m6 = ½(180o) = 90o

3 O

B

4 1

2

9 6

78

5

10 A

C

D

E

80o

20o

50o

100o

110o

19. m7 = ½(100o) = 50o 20. m8 = 20o

21. m9 = ½(50o) = 25o 22. m10 = ½(150o) = 75o