3 APPLY
GUIDED PRACTICE
Concept Check
ASSIGNMENT GUIDE
BASIC
Skill Check
Day 1: pp. 624–627 Exs. 8–34, 40,
42, 43, 46–51
✓
✓
1. If a chord of a circle intersects a tangent to the circle at the point of tangency,
what is the relationship between the angles formed and the intercepted arcs?
The measure of each ™ is half the measure of the intercepted arc.
Find the indicated measure or value.
២
2. mSTU
AVERAGE
3. m™1 60°
210°
U
Day 1: pp. 624–627 Exs. 8–35, 40,
42, 43, 46–51
D
55ⴗ
Day 1: pp. 624–627 Exs. 8–35,
37–40, 42–51
A
60ⴗ
65ⴗ
T
BLOCK SCHEDULE WITH 10.5
B
R
5. m™RQU 90°
pp. 624–627 Exs. 8–35, 40, 42,
43, 46–51
6. m™N 22.5°
7. m™1 88°
92ⴗ
125ⴗ
270ⴗ
U
1
35ⴗ
q
R
88ⴗ
80ⴗ
90ⴗ
N
120ⴗ
92ⴗ
88ⴗ
PRACTICE AND APPLICATIONS
STUDENT HELP
HOMEWORK CHECK
To quickly check student understanding of key concepts, go
over the following exercises:
Exs. 8, 16, 18, 28, 30, 40. See also
the Daily Homework Quiz:
• Blackline Master (Chapter 10
Resource Book, p. 66)
•
Transparency (p. 75)
190ⴗ
1
105ⴗ S
ADVANCED
EXERCISE LEVELS
Level A: Easier
8–13
Level B: More Difficult
14–37, 40, 42, 43
Level C: Most Difficult
38, 39, 41, 44, 45
4. m™DBR 65°
Extra Practice
to help you master
skills is on p. 822.
FINDING MEASURES Find the indicated measure.
8. m™1
២
9. mGHJ
110°
10. m™2 90°
280°
H
G
1
140ⴗ
2
220ⴗ
J
៣
11. mDE
២
12. mABC
72°
252°
13. m™3 110°
A
D
140ⴗ
B
54ⴗ
C
36ⴗ
3
E
xy USING ALGEBRA Find the value of x.
៣
STUDENT HELP
HOMEWORK HELP
Example 1: Exs. 8–13
Example 2: Exs. 14–16
Example 3: Exs. 17–25
Example 4: Exs. 26–28
Example 5: Ex. 35
14. mAB = x° 168
624
២
18
16. mHJK = (10x + 50)°
J
P
A
96ⴗ
Chapter 10 Circles
B
R
(8x ⴚ 29)ⴗ
72ⴗ
H
C
624
៣
15. mPQ = (5x + 17)°
25.4
q
K
FINDING ANGLE MEASURES Find m™1.
17.
1
COMMON ERROR
EXERCISES 17–25 Students
may forget whether they need to
find one-half the sum or the difference to find m⬔1. Refer students
back to the theorems on page 622.
Also, remind students that if the
angle is inside the circle they need
to find the sum and if it is outside
the circle they need to find the
difference.
19. 103°
18. 50°
112.5°
130
1
25
32
75
122
1
95
20. 27°
21. 26°
1
22.
45°
122
105
142
51
70
52
1
1
23. 37°
1
24. 55°
25.
55°
APPLICATION NOTE
Additional information about
fireworks is available at
www.mcdougallittell.com.
1
46
235
125
1
120
xy USING ALGEBRA Find the value of a.
26.
15
260
27.
5
255
28. (a 70)
40
(a 30)
(8a 10)
a
2
15a
FINDING ANGLE MEASURES Use the
diagram at the right to find the
measure of the angle.
INT
STUDENT HELP
NE
ER T
APPLICATION LINK
Visit our Web site
www.mcdougallittell.com.
29. m™1 60°
30. m™2 60°
31. m™3 30°
32. m™4 90°
33. m™5 30°
34. m™6 60°
35.
FIREWORKS You are watching fireworks
over San Diego Bay S as you sail away in a
boat. The highest point the fireworks reach F is
about 0.2 mile above the bay and your eyes E
are about 0.01 mile above the water. At point B
you can no longer see the fireworks because of
the curvature of Earth. The radius of Earth is
Æ
about 4000 miles and FE is tangent to Earth
at T. Find mSB . Give your answer to the
nearest tenth of a degree. 0.7°
៣
1
120
5
3
2
4
60
6
120
F
S
T
E
B
C
Not drawn to scale
10.4 Other Angle Relationships in Circles
625
625
STUDENT HELP NOTES
Software Help Instructions for
several software packages are
available in blackline format in the
Chapter 10 Resource Book, p. 55
and at www.mcdougallittell.com.
41. See Additional Answers beginning on page AA1.
INT
STUDENT HELP
NE
ER T
36.
SOFTWARE HELP
Visit our Web site
www.mcdougallittell.com
to see instructions for
several software
applications.
៣
1
36. m™BAC = }}mAC ;
2
Thm. 10.12
Use geometry software to construct and label circle O,
AB which is tangent to ›O at point A, and any point C on ›O. Then
Æ
construct secant AC. Measure ™BAC and AC . Compare the measures of
™BAC and its intercepted arc as you drag point C on the circle. What do you
notice? What theorem from this lesson have you illustrated? See margin.
TECHNOLOGY
Æ
៣
PROVING THEOREM 10.12 The proof of Theorem 10.12 can be split into
37–39. See margin.
three cases, as shown in the diagrams.
A
A
37. Diameter; 90°; a
tangent line is fi to the
radius drawn to the
point of tangency.
B
B
C
q
A
២
៣
Case 2
The center of the circle is
in the interior of ™ABC.
Case 1
The center of the circle is
on one side of ™ABC.
Æ
37. In Case 1, what type of chord is BC? What is the measure of ™ABC? What
theorem earlier in this chapter supports your conclusion?
38. Write a plan for a proof of Case 2 of Theorem 10.12. (Hint: Use the auxiliary
so by the Arc Addition
Post., m™ABC =
39. Describe how the proof of Case 3 of Theorem 10.12 is different from the
២
line and the Angle Addition Postulate.)
proof of Case 2.
40.
39. The proof would be
similar, using the
Angle Addition and Arc
Addition Postulates,
but you would be
subtracting m™PBC
and mPC instead of
adding.
PROVING THEOREM 10.13 Fill in the blanks
to complete the proof of Theorem 10.13.
៣
Statements
Æ
?㛭㛭㛭
3. m™1 = m™DBC + m™㛭㛭㛭㛭㛭
ACB
1
4. m™DBC = ᎏᎏmDC
2
1
5. m™ACB = ᎏᎏmAB
2
1
1
6. m™1 = ᎏᎏmDC + ᎏᎏmAB
2
2
1
7. m™1 = ᎏᎏ(mDC + mAB )
2
៣
៣
៣ ៣
៣ ៣
ADDITIONAL PRACTICE
AND RETEACHING
For Lesson 10.4:
626
41.
Chapter 10 Circles
B
C
Reasons
Æ
2. Draw BC.
Search the Test and Practice
Generator for key words or
specific lessons.
1
៣
Æ
•
A
1
PROVE 䉴 m™1 = ᎏᎏ(mDC + mAB )
2
1. Chords AC and BD intersect.
626
D
GIVEN 䉴 Chords 苶
AC
苶 and 苶
BD
苶 intersect.
៣
For more Mixed Review:
Case 3
The center of the circle is
in the exterior of ™ABC.
1
1
}}(180°) = }}mBXP ,
2
2
1
and m™QBC =}}mPC ,
2
1
}} m BPC .
2
• Practice Levels A, B, and C
(Chapter 10 Resource Book,
p. 56)
• Reteaching with Practice
(Chapter 10 Resource Book,
p. 59)
•
See Lesson 10.4 of the
Personal Student Tutor
P
q
C
Æ
38. Draw BQ intersecting
the › at P and let X be
a point on the upper
semicircle. Show that
m™ABC = m™ABQ +
m™QBC. Then
m™ABQ = 90° =
B
P
q
C
?㛭㛭㛭 Given.
1. 㛭㛭㛭㛭㛭
?㛭㛭㛭 Through any 2 pts. there is
2. 㛭㛭㛭㛭㛭
exactly 1 line.
?㛭㛭㛭 Ext. ™ Thm.
3. 㛭㛭㛭㛭㛭
Measure of an
?㛭㛭㛭
4. 㛭㛭㛭㛭㛭
inscribed ™ Thm.
?㛭㛭㛭 Measure of an
5. 㛭㛭㛭㛭㛭
inscribed ™ Thm.
?㛭㛭㛭 Substitution Prop.
6. 㛭㛭㛭㛭㛭
?㛭㛭㛭 Distributive Prop.
7. 㛭㛭㛭㛭㛭
JUSTIFYING THEOREM 10.14 Look back at the diagrams for Theorem
10.14 on page 622. Copy the diagram for the case of a tangent and a secant
Æ
and draw BC. Explain how to use the Exterior Angle Theorem in the proof of
this case. Then copy the diagrams for the other two cases, draw appropriate
auxiliary segments, and write plans for the proofs of the cases. See margin.
Test
Preparation
42. MULTIPLE CHOICE The diagram at the right is not
Æ
¡
D
¡
A
¡
E
¡
x < 90
B
x > 90
x ≤ 90
¡
C
l
B
drawn to scale. AB is any chord of the circle. The
line is tangent to the circle at point A. Which
of the following must be true? E
xⴗ
DAILY HOMEWORK QUIZ
A
x = 90
Cannot be determined from given information
43. MULTIPLE CHOICE In the figure at the right, which relationship is not true? C
៣ ៣
៣ º mCD
៣)
¡ m™1 = ᎏ12ᎏ(mEF
៣ º mAC
៣)
¡ m™2 = ᎏ12ᎏ(mBD
៣ º mCD
៣)
¡ m™3 = ᎏ12ᎏ(mEF
A
¡
1
2
m™1 = ᎏᎏ(mCD + mAB )
J
C
A
2
3
(9x + 9)°
1
H
B
C
F
11
2.
PROOF Use the plan to write a paragraph proof.
10(x ⫹ 4)°
See margin.
inscribed in ¤QRS. T, U, and V are points
of tangency.
U
1
2
Plan for Proof Prove that TPVR is a square. Then
Æ
EXTRA CHALLENGE
P
r
T
EXTRA CHALLENGE NOTE
Æ
r
show that QT £ QU and SU £ SV. Finally, use the
Segment Addition Postulate and substitution.
www.mcdougallittell.com
V
R
S
45. FINDING A RADIUS Use the result from Exercise 44 to find the radius of an
inscribed circle of a right triangle with side lengths of 3, 4, and 5. 1
P
USING SIMILAR TRIANGLES Use the diagram at the
right and the given information. (Review 9.1)
20
? 6
47. LM = 4, LN = 9, LP = 㛭㛭㛭
L
Challenge problems for
Lesson 10.4 are available in
blackline format in the Chapter 10
Resource Book, p. 63 and at
www.mcdougallittell.com.
ADDITIONAL TEST
PREPARATION
2. OPEN ENDED Draw a tangent
and a chord intersecting at a
point on a circle. Estimate the
angle between them. Label each
angle and intercepted arc with
its measure. Sample answer:
MIXED REVIEW
?
46. MN = 9, PM = 12, LP = 㛭㛭㛭
(5x ⫺ 4)°
5
PROVE 䉴 r = ᎏᎏ(QR + RS º QS)
Æ
(9x ⫹ 3)°
q
GIVEN 䉴 ™R is a right angle. Circle P is
Æ
K
D
D
44.
Transparency Available
Find the value of x.
២ = (20x – 4)°
1. mHJK
E
B
★ Challenge
4 ASSESS
M
N
240°
48. FINDING A RADIUS You are 10 feet from a circular storage tank. You
are 22 feet from a point of tangency on the tank. Find the tank’s radius.
120°
19.2 ft
(Review 10.1)
60°
120°
Æ
Æ
xy USING ALGEBRA AB
and AD are tangent to ›L. Find the value of x.
(Review 10.1)
49. 25
50.
8
B
x
A
2x ⴚ 5
L
2
51.
B
44. See Additional Answers beginning on page AA1.
B
6x ⴙ 12
L
L
25
D
D
xⴙ3
A
A
10x ⴙ 4
D
10.4 Other Angle Relationships in Circles
627
627