11.2 Central Angles, Inscribed Angles, and Intercepted Arcs
Bell Work:
1. Draw a circle P and label a semi-circle ABC.
2. Draw chords AB and BC to form <ABC.
Is this angle acute, right or obtuse? Right
A
P
Read the paragraph at the top of page 850.
Recall the degree measure of a circle is 360°.
Therefore the degree measure of a semicircle is 180°.
B
Each minor arc of a circle is associated with and determined by a specific central angle.
The degree measure of a minor arc is the same as the degree measure of its
central angle.
If <PRO is a central angle and m<PRO = 30°, then mPO = 30°.
P
R
O
With your partner complete problem 1 questions 1acef on page 850 and 851.
1a. <AOB and <CPD
1b. m<AOB = 120° and m<CPD = 60°
1c. m<AOB = 2m<CPD
1e. mAB = 120° and mCD = 60°
1f.
mAZB = 360° – 120° = 240°
C
D
A
B
mCZD = 360° – 60° = 300°
Z
C
Z
Read the sentence at the top of page 852.
Adjacent arcs are two arcs of the same circle sharing a common endpoint.
With your partner complete problem 1 question 4 on page 852.
XY and YZ are adjacent arcs.
Y
X
X
Read the sentence below problem 1 question 4 on page 852.
Arc Addition Postulate
The measure of an arc formed by two adjacent arcs in the sum of the
measures of the two arcs.
mAB + mBC = mABC
With your partner complete problem 1 question 5 on page 852.
mXY + mYZ = mXYZ
Read the paragraph below problem 1 question 5 on page 852.
B
A
C
An intercepted arc is an arc associated with and determined by angles of the circle.
An intercepted arc is a portion of the circumference of the circle located on the interior of the angle
whose endpoints lie on the sides of an angle.
R
With your partner complete problem 1 questions 6ab.
a. P
b. PR
O
S
Investigate the relationship between an inscribed angle and a central angle that intercept the same arc.
With your partner complete problem 1 question 7abcd, 10.
7a. What do <AOB and <APB have in common? They intercept the same arc, AB
P
b. m<APB = 45° and m<AOB = 90° so m<APB is ½ m<AOB
Since m<AOB = mAB then m<APB = ½ mAB
P
P
c.
m<AQB = ½ mAB
d.
Q
Q
m<ARB
=
½
mAB
O
R
A
O
O
B
A
B
A
B
Highlight the Inscribed Angle Theorem at the bottom of page 857.
P
Inscribed Angle Theorem
The measure of an inscribed angle is half the measure of its intercepted arc.
mAPB = ½ mAB
(We will not prove this theorem.)
O
A
B
Use the space on page 857 to write the following information.
Definition of congruent arcs
If mAB = mCD, then AB CD.
Note the notation is similar to angles, the m is used with = but not with
.
Theorem
If two inscribed angles intercept the same arc, then the angles are congruent.
A
Given: <ABC and <ADC
Prove: <ABC <ADC
Strategy: Recall each of these angles is ½ their intercepted arc.
With the class complete the proof.
B
C
D
Proof:
Statements
1. m<ABC = ½ mAC
2. m<ADC = ½ mAC
3. m<ABC = m<ADC
4. <ABC <ADC
Reasons
1.
2.
3.
4.
Definition of measure of an inscribed angle
Definition of measure of an inscribed angle
Substitution Property
Definition of congruent angles
Highlight the Parallel Lines-Congruent Arcs Theorem at the bottom of page 859.
Parallel Lines-Congruent Arcs Theorem
Parallel lines intercept congruent arcs on a circle.
O
A
Use the picture and space provided on page 859 to complete this proof
P
with the class.
R
L
Given: AP||LR
Prove: PL AR
Strategy: Draw chord PR. PR is a transversal making <APR and <LRP alternate interior angles, therefore
congruent. Recall each of these angles is ½ their intercepted arcs.
Proof:
Statements
Reasons
1.
2.
3.
4.
5.
6.
7.
8.
AP||LR
<APR <LRP
m<APR = m<LRP
m<APR = ½ mAR
m<LRP = ½ mLP
½ mAR = ½ mLP
mAR = mLP
AR LP
1.
2.
3.
4.
5.
6.
7.
8.
Given
Alternate Interior Angle Theorem
Definition of congruent angles
Definition of measure of an inscribed angle
Definition of measure of an inscribed angle
Substitution Property
Multiplication Property of Equality
Definition of congruent arcs
Homework:
Skills Practice Problem Set 11.2 prob. 2, 3, 8, 9, 14, 15, 20, 21, 26, 27, 33, 34.