9.6 Secants, Tangents and
Angle Measures
Geometry
Objectives
• Use angles formed by tangents and
chords to solve problems in
geometry.
• Use angles formed by lines that
intersect a circle to solve problems.
Using Tangents and Chords
• Measure of an
angle inscribed in
a circle is half the
measure of its
intercepted arc.
A
C
D
B
m ADB = ½m AB
Theorem 9.11
• 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.
B
C
1
2
A
m1= ½m AB
m2= ½m ABC
Finding Angle and Arc
Measures
m
• Line m is tangent to
the circle. Find the
measure of the red
angle or arc.
• Solution:
m1= ½ AB
m1= ½ (150°)
m1= 75°
B
1
A
150°
Finding Angle and Arc
Measures
S
• Line m is tangent to
the circle. Find the
measure of the red
angle or arc.
130°
P
• Solution:
m RSP = 2(130°)
m RSP = 260°
R
Finding an Angle Measure
BC is tangent to the
circle. Find m CBD (9x + 20)°
• Solution:
m CBD = ½ m DAB
5x = ½(9x + 20)
10x = 9x +20
x = 20
D
mCBD = 5(20°) = 100°
C
A
5x°
B
D
m1 = ½ ( m CD + m AB)
1
A
2
B
C
m2 = ½ ( m BC+ m AD)
Finding the Measure of an Angle
Formed by Two Chords
106°
P
• Find the value of x
• Solution:
x° = ½ (m QR +m PS)
x° = ½ (106° + 174°)
x = 140
S
Q
x°
R
174°
E
Using Theorem 9.13
200°
• Find the value of x
D
F
m GHF = ½ (m EGD - m GF )
72° = ½ (200° - x°)
144 = 200 - x°
- 56 = -x
56 = x
x°
G
H
72°
Using Theorem 9.13
M
Because MN and MLN make a whole
circle, m MLN =360°-92°=268°
L
92°
• Find the value of x
m GHF = ½ (m MLN - m MN)
= ½ (268 - 92)
= ½ (176)
= 88
N
x°
P
Practice
Practice
m1 = ½ ( 40 + 52) =46
m2 = ½ ( 134) = 67
m3 = ½ ( 100 – 70) = 15
Practice
100 = ½ ( 130 + x)
200 = 130+ x
X = 70
50 = ½ ( (360 – x) -x)
100 = 360- 2x
260 = 2x
X = 130
20 = ½ ( 70 – x)
40 = 70-x
X = 30
CD = CQD = 120
 E = ½ ( AD -BC)
25 = ½ (x -30)
50 = x – 30
X = 80
AB = 360-30 – 120 – 80 =
AB = 130
QDC = (180- 120) / 2 = 30
360 = 140+ 2y + y +2y
360= 140 +5y
220 = 5y
Y = 44
Y = 44
2 * 44 = 88
Y = 44
2 * 44 = 88
BCD = ½( AE – BD)
BCD = ½( 140-44)
BCD = 48
A = FB = 50
BCA = ½ * FB = 25
ABC = 180- 50 -25 = 105
GBC =180-105 =75
360 = 4x – 50 +x + x + 25+ x – 15 + 50
360=7x +10
350 = 7x
X = 50
FHE = ½( 35 + 50)
FHE = 42.5
X = 50
CFD = ½*50 = 25