Download Section 10.6

T
O
P
O
B
S
A
C
More Angle-Arc Theorems
X
A
Section
10.6
B
A
P
C
B
Y
D
Objectives
• Recognize congruent inscribed and tangent-chord
angles
• Determine the measure of an angle inscribed in
a semicircle
• Apply the relationship between the measures of
a tangent-tangent angle and its minor arc
InCongruent Inscribed and
Tangent-Chord Angles
–Theorem 89:
• If two inscribed or tangent-chord angles intercept
the same arc, then they are congruent
X
A
B
Y
InCongruent Inscribed and
Tangent-Chord Angles
Given: X and Y are inscribed angles
intercepting arc AB
X
Conclusion:
X

Y
A
B
Y
Congruent Inscribed and
tangent Chord Angles
–Theorem 90:
• If two inscribed or tangent-chord angles intercept
congruent arcs, then they are congruent
A
P
C
E
B
D
Congruent Inscribed and
tangent Chord Angles
If ED is the tangent at D and AB CD, we
may conclude that P
CDE.
A
P
C
E
B
D
Angles Inscribed in
Semicircles
–Theorem 91:
• An angle inscribed in a semi circle is a right angle
O
B
A
C
A Special Theorem About
Tangent-Tangent Angles
–Theorem 92:
• The sum of the measure of a tangent and its minor
arc is 180o
T
O
P
S
A Special Theorem About
Tangent-Tangent Angles
Given: PT and
are tangent to circle O.
PS
Prove: m P + mTS = 180
T
O
P
S
A Special Theorem About
Tangent-Tangent Angles
Proof: Since the sum of the measures of the angles
in quadrilateral SOTP is 360 and since T and
S are right angles, m P + m O = 180.
T
O
Therefore, m
P
P + mTS = 180.
S
T
O
P
O
B
S
A
C
Sample Problems
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A
A
P
B
C
B
Y
D
Problem 1
V
S
Given: ʘO
E
O
L
N
Conclusion:
∆LVE ∆NSE,
EV ● EN = EL ● SE
V
S
Problem 1 - Proof
1. ʘO
2. V  S
L
1. Given
2. If two inscribed s
intercept the same arc, they

are .
3. L  N
3. Same as 2
4. ∆LVE ∆NSE
4. AA (2,3)
5. EV = EL
5. Ratios of corresponding
SE
EN
sides of ~ ∆ are =.s
6. EV●EN = EL●SE
6. Means-Extremes Products
Theorem
E
O
N
Problem 2
In Circle O, BCis a diameter
and the radius of the circle is 20.5 mm.
Chord
AChas a length of 40 mm.
A
Find AB.
B
C
O
Problem 2 - Solution
Since A is inscribed in a semicircle, it is a right
angle. By the Pythagorean Theorem,
(AB)2 + (AC)2 = (BC)2
A
(AB)2 + 402 = 412
C
B
AB = 9 mm
O
Problem 3
Given: ʘO with



‖
CD
AB
Prove:
C


AB
tangent at B,
BDC
B
A
C
D
O
N
Problem 3 - Proof
AB
1. 
is tangent to ʘO.
CD
2. 
‖ 
AB
3. ABD  BDC
4. C  ABD
A
B
D
O
5.
C
BDC
1. Given
2. Given
3. ‖ lines ⇒alt. int. s
4. If an inscribed and a
tangent-chord
C
intercept the same
arc, they are . 
N
5. Transitive Property

T
O
P
O
B
S
A
C
You try!!
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A
A
P
B
C
B
Y
D
Problem A
Given: PQand
QR = 163o
are tangent segments.
PR
Q
Find:
a.
b.
P
PQR
P
R
Solution
a. 163o + P = 180o
P = 17o
b. PQR  PRQ (intercept the same arc)
17o + 2x = 180o
2x = 163o
x = 81.5o
PQR = 81.5o
Q
P
R
Problem B
Given: A, B, and C are points of contact.
X
o
o
AB = 145 , Y = 48
Find:
Z
A
B
Y
Z
C
Solution
X
X + 145o = 180o
X = 35o
X + Y + Z = 180o
35o + 48o + Z = 180o
Z = 97o
A
B
Y
Z
C
Problem C
In the figure shown, find m
P.
A
P
(6x)o
(15x + 33)o
B
Solution
P + AB = 180o
6x + 15x + 33o = 180o
21x = 147o
x = 7o
P (6x)
m P = 6x
m P = 42o
o
A
(15x + 33)o
B