More Angle-Arc Theorems
Section 10.6X
Y
A
B P
C
A
B
D
OB A
C
O
T
P
S
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
Y
A
B
InCongruent Inscribed and Tangent-Chord
Angles
Given: X and Y are inscribed angles intercepting arc AB
Conclusion: X Y
X
Y
A
B
Congruent Inscribed and tangent Chord Angles
–Theorem 90:• If two inscribed or tangent-chord angles intercept congruent arcs, then they are congruentP
C
A
B
DE
Congruent Inscribed and tangent Chord Angles
If is the tangent at D and AB CD, we may conclude that P CDE.
P
C
A
B
DE
ED
Angles Inscribed in Semicircles
–Theorem 91:• An angle inscribed in a semi circle is a right angle
OB 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
O
T
P
S
A Special Theorem About Tangent-Tangent Angles
Given: and are tangent to circle O.
Prove: m P + mTS = 180
O
T
P
S
PT PS
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.
Therefore, m P + mTS = 180.
O
T
P
S
Sample ProblemsX
Y
A
B P
C
A
B
D
OB A
C
O
T
P
S
Problem 1
Given: ʘO
Conclusion: ∆LVE ∆NSE,
EV ● EN = EL ● SE
VS
N
L
O
E
Problem 1 - Proof1. ʘO 1. Given2. V S 2. If two inscribed s
intercept the same arc, they are .
3. L N 3. Same as 24. ∆LVE ∆NSE 4. AA (2,3)
5. = 5. Ratios of corresponding sides of ~ ∆ are =.
6. EV●EN = EL●SE 6. Means-Extremes Products Theorem
V S
N
LO
E
SE
EV
EN
EL
s
Problem 2
In Circle O, is a diameterand the radius of the circle is 20.5
mm.
Chord has a length of 40 mm.
Find AB. O
A
CB
BC
AC
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
(AB)2 + 402 = 412
AB = 9 mmO
A
CB
Problem 3
Given: ʘO with tangent at B, ‖
Prove: C BDC
A B
N
D
O
C
AB
AB
CD
Problem 3 - Proof1. is tangent to ʘO. 1. Given2. ‖ 2. Given3. ABD BDC 3. ‖ lines ⇒alt. int. s4. C ABD 4. If an inscribed and a tangent-chord
intercept the same arc, they are .
5. C BDC 5. Transitive Property
AB
AB
CD
A B
N
DO
C
You try!!X
Y
A
B P
C
A
B
D
OB A
C
O
T
P
S
Problem A
Given: and are tangent segments.QR = 163o
Find: a. P b. PQR
Q
P
R
PQ PR
Solutiona. 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.AB = 145o, Y = 48o
Find: Z B
X
CYZ
A
Solution
X + 145o = 180o X = 35o
X + Y + Z = 180o
35o + 48o + Z = 180o
Z = 97o
B
X
CYZ
A
Problem C
In the figure shown, find m P.
A
P
B
(6x)o (15x + 33)o
Solution
P + AB = 180o 6x + 15x + 33o = 180o
21x = 147o
x = 7o
m P = 6x m P = 42o
A
P
B
(6x)o
(15x + 33)o