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