Section 10.4 Inscribed Angles and Polygons 553
Inscribed Angles and Polygons
Essential QuestionEssential Question How are inscribed angles related to their
intercepted arcs? How are the angles of an inscribed quadrilateral related to
each other?
An inscribed angle is an angle whose
vertex is on a circle and whose sides
contain chords of the circle. An arc that
lies between two lines, rays, or segments is
called an intercepted arc. A polygon is an
inscribed polygon when all its vertices lie
on a circle.
Inscribed Angles and Central Angles
Work with a partner. Use dynamic geometry software.
a. Construct an inscribed angle
in a circle. Then construct the
corresponding central angle.
b. Measure both angles. How is
the inscribed angle related to
its intercepted arc?
c. Repeat parts (a) and (b) several
times. Record your results in a
table. Write a conjecture about
how an inscribed angle is related
to its intercepted arc.
A Quadrilateral with Inscribed Angles
Work with a partner. Use dynamic geometry software.
a. Construct a quadrilateral with
each vertex on a circle.
b. Measure all four angles. What
relationships do you notice?
c. Repeat parts (a) and (b) several
times. Record your results in a
table. Then write a conjecture
that summarizes the data.
Communicate Your AnswerCommunicate Your Answer 3. How are inscribed angles related to their intercepted arcs? How are the angles of
an inscribed quadrilateral related to each other?
4. Quadrilateral EFGH is inscribed in ⊙C, and m∠E = 80°. What is m∠G? Explain.
ATTENDING TO PRECISION
To be profi cient in math, you need to communicate precisely with others.
10.4
R
Q
P
O
central angle
intercepted arc
Rinscribed angle
A
C
D
B
Sample
A
C
D
E
B
Sample
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554 Chapter 10 Circles
10.4 Lesson What You Will LearnWhat You Will Learn Use inscribed angles.
Use inscribed polygons.
Using Inscribed Angles
The proof of the Measure of an Inscribed Angle Theorem involves three cases.
CC C
Case 1 Center C is on a
side of the inscribed angle.
Case 2 Center C is
inside the inscribed angle.
Case 3 Center C is
outside the inscribed angle.
Using Inscribed Angles
Find the indicated measure.
a. m∠T
b. m � QR
SOLUTION
a. m∠T = 1 —
2 m � RS =
1 —
2 (48°) = 24°
b. m � TQ = 2m∠R = 2 ⋅ 50° = 100° Because � TQR is a semicircle, m � QR = 180° − m � TQ = 180° − 100° = 80°.
inscribed angle, p. 554intercepted arc, p. 554subtend, p. 554inscribed polygon, p. 556circ*mscribed circle, p. 556
Core VocabularyCore Vocabullarry
Core Core ConceptConceptInscribed Angle and Intercepted ArcAn inscribed angle is an angle whose
vertex is on a circle and whose sides contain
chords of the circle. An arc that lies between
two lines, rays, or segments is called an
intercepted arc. If the endpoints of a chord
or arc lie on the sides of an inscribed angle,
then the chord or arc is said to subtend
the angle.
TheoremTheoremTheorem 10.10 Measure of an Inscribed Angle TheoremThe measure of an inscribed angle is one-half
the measure of its intercepted arc.
Proof Ex. 37, p. 560
A
C
B interceptedarc
inscribedangle
A
C
B
D
R
Q
P
ST
50°48°
∠B intercepts � AC . � AC subtends ∠B.AC subtends ∠B.
m∠ADB = 1 — 2 m � AB
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Section 10.4 Inscribed Angles and Polygons 555
Finding the Measure of an Angle
Given m∠E = 75°, fi nd m∠F.
SOLUTION
Both ∠E and ∠F intercept � GH . So, ∠E ≅ ∠F by the Inscribed Angles of a
CircleTheorem.
So, m∠F = m∠E = 75°.
Monitoring ProgressMonitoring Progress Help in English and Spanish at BigIdeasMath.com
Find the measure of the red arc or angle.
1.
G
H
D
F
90°
2. T U
V
38° 3. X
WZ
Y
72°
Finding the Measure of an Intercepted Arc
Find m � RS and m∠STR. What do you notice
about ∠STR and ∠RUS?
SOLUTION
From the Measure of an Inscribed Angle Theorem, you know that
m � RS = 2m∠RUS = 2(31°) = 62°.
Also, m∠STR = 1 —
2 m � RS =
1 —
2 (62°) = 31°.
So, ∠STR ≅∠RUS.
Example 2 suggests the Inscribed Angles of a Circle Theorem.
TheoremTheoremTheorem 10.11 Inscribed Angles of a Circle TheoremIf two inscribed angles of a circle intercept
the same arc, then the angles are congruent.
Proof Ex. 38, p. 560
TS
UR31°
A
BC
D
G
HF
E 75°
∠ADB ≅ ∠ACB
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556 Chapter 10 Circles
Using Inscribed Polygons
Find the value of each variable.
a.
A
B
Q
C
2x°
b.
G
ED
F120°
80°z°
y°
SOLUTION
a. — AB is a diameter. So, ∠C is a right angle, and m∠C = 90° by the Inscribed Right
Triangle Theorem.
2x° = 90°
x = 45
The value of x is 45.
b. DEFG is inscribed in a circle, so opposite angles are supplementary by the
Inscribed Quadrilateral Theorem.
m∠D + m∠F = 180° m∠E + m∠G = 180°
z + 80 = 180 120 + y = 180
z = 100 y = 60
The value of z is 100 and the value of y is 60.
Using Inscribed Polygons
Core Core ConceptConceptInscribed PolygonA polygon is an inscribed polygon when all
its vertices lie on a circle. The circle that
contains the vertices is a circ*mscribed circle.
TheoremsTheoremsTheorem 10.12 Inscribed Right Triangle TheoremIf a right triangle is inscribed in a circle, then the hypotenuse
is a diameter of the circle. Conversely, if one side of an
inscribed triangle is a diameter of the circle, then the triangle
is a right triangle and the angle opposite the diameter is the
right angle.
Proof Ex. 39, p. 560
Theorem 10.13 Inscribed Quadrilateral TheoremA quadrilateral can be inscribed in a circle if and only if its
opposite angles are supplementary.
Proof Ex. 40, p. 560;
BigIdeasMath.com
inscribedpolygon
circ*mscribedcircle
B
A
D
C
D
EF
CG
m∠ABC = 90° if and only if AC is a diameter of the circle.
D, E, F, and G lie on ⊙C if and only if m∠D + m∠F = m∠E + m∠G = 180°.
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Section 10.4 Inscribed Angles and Polygons 557
Using a Circ*mscribed Circle
Your camera has a 90° fi eld of vision, and you want to
photograph the front of a statue. You stand at a location
in which the front of the statue is all that appears in
your camera’s fi eld of vision, as shown. You want to
change your location. Where else can you stand so
that the front of the statue is all that appears inyour
camera’s fi eld of vision?
SOLUTION
From the Inscribed Right Triangle Theorem, you know
that if a right triangle is inscribed in a circle, then the
hypotenuse of the triangle is a diameter of the circle. So,
draw the circle that has the front of the statue as
a diameter.
The statue fi ts perfectly within your camera’s
90° fi eld of vision from any point on the
semicircle in front of the statue.
Monitoring ProgressMonitoring Progress Help in English and Spanish at BigIdeasMath.com
Find the value of each variable.
4.
K
ML40°x°
y°
5.
A
C
D
B68°
82°
x°
y°
6.
V
T
U
S10x°
8x°
(2c − 6)°c°
7. In Example 5, explain how to fi nd locations where the left side of the statue
is all that appears in your camera’s fi eld of vision.
Step 1 Step 2 Step 3
A
B
C
A
B
CE D
A
B
CE D
Draw a diameter Draw any diameter. Label the
endpoints A and B.
Construct a perpendicular bisector
Construct the perpendicular bisector
of the diameter. Label the points
where it intersects ⊙C as points D
and E.
Form a square Connect points A, D, B, and E to
form a square.
Constructing a Square Inscribed in a Circle
Given ⊙C, construct a square inscribed in a circle.
SOLUTION
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558 Chapter 10 Circles
Exercises10.4 Dynamic Solutions available at BigIdeasMath.com
1. VOCABULARY If a circle is circ*mscribed about a polygon, then the polygon is an _______________.
2. DIFFERENT WORDS, SAME QUESTION Which is different?
A
E
C
DG
F
B
25°25°
Find “both” answers.
Find m∠ABC.
Find m∠AEC.
Find m∠AGC.
Find m∠ADC.
Vocabulary and Core Concept CheckVocabulary and Core Concept Check
In Exercises 3–8, fi nd the indicated measure. (See Examples 1 and 2.)
3. m∠A 4. m∠G
A
C
B
84°
F
D
G
70°
120°
5. m∠N 6. m � RS
L
N
M
160°
R
Q
S
67°
7. m � VU 8. m � WX
U
T
V
30°
Y
W
X
75° 110°
In Exercises 9 and 10, name two pairs of congruent angles.
9.
C
A
B
D
10.
ZX
W
Y
In Exercises 11 and 12, fi nd the measure of the red arc or angle. (See Example 3.)
11. F
H
E
G
51°
12. Q
S
P
R40°
45°
In Exercises 13–16, fi nd the value of each variable. (See Example 4.)
13.
Q
S
T
R
95°80°
x° y° 14. E F
GD60° 60°
m° 2k°
15. J
K
L
M
110°
130°
54°4b°
3a°
16. Y
Z
X
34° 2y°3x°
17. ERROR ANALYSIS Describe and correct the error in
fi nding m � BC .
m � BC = 53°
✗ B
C
A 53°
Monitoring Progress and Modeling with MathematicsMonitoring Progress and Modeling with Mathematics
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Section 10.4 Inscribed Angles and Polygons 559
18. MODELING WITH MATHEMATICS A carpenter’s square is an L-shaped tool used to draw right angles.
You need to cut a circular piece of wood into two
semicircles. How can you use the carpenter’s square
to draw a diameter on the circular piece of wood? (See Example 5.)
MATHEMATICAL CONNECTIONS In Exercises 19–21, fi nd the values of x and y. Then fi nd the measures of the interior angles of the polygon.
19. B
C
D
A 3x°26y°
21y°
2x°
20.
B
C
D
A
14x°
24y°9y°
4x°
21. B
C
A 2x°
6y°
6y°
4x°
22. MAKING AN ARGUMENT Your friend claims that
∠PTQ ≅ ∠PSQ ≅ ∠PRQ. Is your friend correct?
Explain your reasoning.
P
QR
S
T
23. CONSTRUCTION Construct an equilateral triangle
inscribed in a circle.
24. CONSTRUCTION The side length of an inscribed
regular hexagon is equal to the radius of the
circ*mscribed circle. Use this fact to construct
aregular hexagon inscribed in a circle.
REASONING In Exercises 25–30, determine whether a quadrilateral of the given type can always be inscribed inside a circle. Explain your reasoning.
25. square 26. rectangle
27. parallelogram 28. kite
29. rhombus 30. isosceles trapezoid
31. MODELING WITH MATHEMATICS Three moons,
A, B, and C, are in the same circular orbit
100,000 kilometers above the surface of a planet.
The planet is 20,000 kilometers in diameter and
m∠ABC = 90°. Draw a diagram of the situation.
How far is moon A from moon C?
32. MODELING WITH MATHEMATICS At the movie
theater, you want to choose a seat that has the best
viewing angle, so that you can be close to the screen
and still see the whole screen without moving your
eyes. You previously decided that seat F7 has the best
viewing angle, but this time someone else is already
sitting there. Where else can you sit so that your seat
has the same viewing angle as seat F7? Explain.
movie screen
Row ARow BRow CRow DRow ERow FRow G
F7
33. WRITING A right triangle is inscribed in a circle, and
the radius of the circle is given. Explain how to fi nd
the length of the hypotenuse.
34. HOW DO YOU SEE IT? Let point Y represent your
location on the soccer fi eld below. What type of
angle is ∠AYB if you stand anywhere on the circle
except at point A or point B?
A
B
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560 Chapter 10 Circles
35. WRITING Explain why the diagonals of a rectangle
inscribed in a circle are diameters of the circle.
36. THOUGHT PROVOKING The fi gure shows a circle
that is circ*mscribed about △ABC. Is it possible
tocirc*mscribe a circle about any triangle? Justify
your answer.
C
A
B
37. PROVING A THEOREM If an angle is inscribed in ⊙Q,
the center Q can be on a side of the inscribed angle,
inside the inscribed angle, or outside the inscribed
angle. Prove each case of the Measure of an Inscribed
Angle Theorem (Theorem 10.10).
a. Case 1
Given ∠ABC is inscribed
in ⊙Q.
Let m∠B = x°. Center Q lies on — BC .
Prove m∠ABC = 1 —
2 m � AC
(Hint: Show that △AQB is isosceles. Then write
m � AC in terms of x.)
b. Case 2 Use the diagram and
auxiliary line to write Given
and Prove statements for
Case 2. Then write a proof.
c. Case 3 Use the diagram and
auxiliary line to write Given
and Prove statements for
Case 3. Then write a proof.
38. PROVING A THEOREM Write a paragraph proof
oftheInscribed Angles of a Circle Theorem
(Theorem10.11). First, draw a diagram and write
Given and Prove statements.
39. PROVING A THEOREM The Inscribed Right Triangle
Theorem (Theorem 10.12) is written as a conditional
statement and its converse. Write a plan for proof for
each statement.
40. PROVING A THEOREM Copy and complete the
paragraph proof for one part of the Inscribed
Quadrilateral Theorem (Theorem 10.13).
Given ⊙C with inscribed
quadrilateral DEFG
Prove m∠D + m∠F = 180°, m∠E + m∠G = 180°
By the Arc Addition Postulate (Postulate 10.1),
m � EFG + ____ = 360° and m � FGD + m � DEF = 360°. Using the ________ Theorem, m � EDG = 2m∠F,
m � EFG = 2m∠D, m � DEF = 2m∠G, and
m � FGD = 2m∠E. By the Substitution Property
ofEquality, 2m∠D + ___ = 360°, so ___.
Similarly, ___.
41. CRITICAL THINKING In the diagram, ∠C is a right
angle. If you draw the smallest possible circle through
C tangent to — AB , the circle will intersect — AC at J and — BC at K. Find the exact length of — JK .
B5
43
A
C
42. CRITICAL THINKING You are
making a circular cutting board.
To begin, you glue eight 1-inch
boards together, as shown.
Then you draw and cut a circle
with an 8-inch diameter from
theboards.
a. — FH is a diameter of the circular cutting board.
Write a proportion relating GJ and JH. State a
theorem to justify your answer.
b. Find FJ, JH, and GJ. What is the length of the
cutting board seam labeled — GK ?
Maintaining Mathematical ProficiencyMaintaining Mathematical ProficiencySolve the equation. Check your solution. (Skills Review Handbook)
43. 3x = 145 44. 1 — 2 x = 63
45. 240 = 2x 46. 75 = 1 —
2 (x − 30)
Reviewing what you learned in previous grades and lessons
C
A
BQ
x°
C
A
BDQ
CA
BDQ
C
E
G
D
F
G
HJF
L
MK
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