Circles—Angles and Arcs
TEACHER NOTES
MATH NSPIRED: GEOMETRY
Math Objectives
•
Students will know the definitions of and identify central angles,
major and minor arcs, intercepted arcs, and inscribed angles of a
circle.
•
Students will determine and apply the following relationships:
•
Two inscribed angles intercepting the same arc have the
same measure.
•
•
An inscribed angle measure of 90° results in the endpoints of
TI-Nspire™ Technology
the intercepted arc lying on a diameter.
Skills:
The measure of an angle inscribed in a circle is half the
• Download a TI-Nspire
measure of the central angle that intercepts the same arc.
•
•
document
Students will construct viable arguments and critique the
• Open a document
reasoning of others (CCSS Mathematical Practice).
• Move between pages
Students will look for and express regularity in repeated
reasoning (CCSS Mathematical Practice).
• Grab and drag a point
• Attach segments to an angle
and an angle to a segment
Vocabulary
•
central angle
Tech Tips:
•
inscribed angle
• Make sure the font size on
•
major, minor, and intercepted arc
your TI-Nspire handheld is
set to Medium.
About the Lesson
•
This lesson involves manipulating endpoints of an arc,
manipulating an inscribed angle, and manipulating the vertex of
an angle intercepting an arc.
•
•
•
Use visualization to understand the definitions of central
Lesson Materials:
Student Activity
Infer that the sum of the measures of minor and major arcs is
Circles_Angles_and_Arcs_
360°, that two inscribed angles intercepting the same arc
Student.pdf
have the same measure, and that the inscribed angle has half
Circles_Angles_and_Arcs_
the measure of the central angle that intercepts the same arc.
Student.doc
Deduce that the opposite angles of a quadrilateral inscribed
Use Screen Capture to examine examples of systems of
inequalities.
Use Teacher Edition computer software or Live Presenter to
review student documents and discuss examples as a class.
©2011 Texas Instruments Incorporated
TI-Nspire document
Circles_Angles_and_Arcs.tns
TI-Nspire™ Navigator™ System
•
the function line in a Graphs
angle, intercepted arc, and minor and major arcs.
in a circle are supplementary.
•
the function line or access
& Geometry page.
As a result students will:
•
• Press / G to either hide
1
Visit www.mathnspired.com for
lesson updates and tech tip
videos.
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Circles—Angles and Arcs
TEACHER NOTES
MATH NSPIRED: GEOMETRY
•
Use Quick Poll to assess student understanding throughout the
lesson.
Discussion Points and Possible Answers
Tech Tip: If students experience difficulty dragging a point, check to make
sure that they have moved the cursor until it becomes a hand (÷) getting
ready to grab the point. Press / x to grab the point and close the
hand ({).
Move to page 1.2.
1. Drag point A or point C. Describe the changes that occur in
the figure as you drag the point.
Answer: The angle measurement changes but is never
more than 180°. The solid part of the circle is always in the
interior of the angle.
Tech Tip: The dashed attribute will not show on the handheld; that part of
the circle will show up as gray instead of black.
2. Angle AOC is called a central angle. Why do you think this is so?
Answer: The vertex is at the center of the circle.
An angle intercepts an arc of a circle if each endpoint of the arc is on a different ray of the angle and the
other points of the arc are in the interior of the angle.
Move to page 1.3.
As you move point A or point C, the central angle ∠AOC intercepts a
minor arc AC. The measure of the minor arc equals the measure of
the central angle. The larger remaining arc, ABC, is called a major
arc.
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Circles—Angles and Arcs
TEACHER NOTES
MATH NSPIRED: GEOMETRY
Teacher Tip: On the TI-Nspire, arcs are measured using Length, not
Angle. Therefore, if a student uses the built-in measuring tool, TI-Nspire
will report arc length rather than arc measure.
3. a. Move point A or point C to help you complete the table.
Sample answer: The completed table is below. The final row of students’ tables will vary.
∠AOC
arc AC
arc ABC
arc AB + arc ABC
50°
50°
310°
360°
100°
100°
260°
360°
110°
110°
250°
360°
(Choose an angle.)
60°
60°
300°
360°
b. What is true about the measure of AB + ABC, the sum of the measures of the minor and major
arcs?
Answer: The measures of AB + ABC always equals 360°.
4. In a circle, the measure of a central angle ∠AOC is n°.
a. What is the measure of the minor arc that is intercepted by the central angle? How do you know?
Answer: The minor arc measures n° because an intercepted arc has the same measurement as
its central angle.
b. What is the measure of the major arc? How do you know?
Answer: The major arc measures (360 – n)° because the sum of the measures of the major and
minor arcs is 360°.
TI-Nspire Navigator Opportunity: Quick Poll
See Note 1 at the end of this lesson.
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Circles—Angles and Arcs
TEACHER NOTES
MATH NSPIRED: GEOMETRY
Move to page 1.4.
5. Angle ABC is called an inscribed angle because BA and BC
are chords of the circle and vertex B is on the circle. Drag point
B around the circle.
a. As point B is moved around the circle, what do you notice about the measure of ∠ABC?
Answer: Angle ABC has the same measure until it intercepts the other arc. While point B is
moved around on that arc, ∠ABC will remain the same.
Teacher Tip: Some students may recognize that the two angles’ measures
sum to 180°.
b. Why does m∠ABC change when point B is moved from one arc to the other? Explain your
reasoning.
Answer: The angle measure changes because the intercepted arc changes. The intercepted arc
is always in the interior of the inscribed angle.
c.
Move point A or point C until ∠ABC is a right angle. What is special about the arc and AC ?
Answer: The arc measure is 180° and the arc is a semicircle. AC is a diameter.
Teacher Tip: When students reach the right angle, the diameter should
show up as a dotted segment.
TI-Nspire Navigator Opportunity: Screen Capture
See Note 2 at the end of this lesson.
Move to page 1.5.
Angle ABC intercepts arc AC. Drag point D to various locations
outside the circle, on the circle, inside the circle, and at the center O.
6. Place point D on the circle so that ∠ADC intercepts the same arc
as ∠ABC.
a. What do you notice about the measures of ∠ABC and ∠ADC?
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Circles—Angles and Arcs
TEACHER NOTES
MATH NSPIRED: GEOMETRY
Answer: The angle measures are the same. The angles are congruent. The ratio of the angle
measurements is 1.
TI-Nspire Navigator Opportunity: Quick Poll
See Note 3 at the end of this lesson.
Teacher Tip: Make sure both angles intercept the same arc. Some
students may incorrectly place point D on the (bold) arc intercepted by
∠ABC. Point D should “snap” to the circle when it gets close.
b. What happens to the angles if you move point A or point C?
Answer: The angle measures change. The angles are congruent and the ratio of the measure of
∠ADC to the measure of ∠ABC is 1 if the angles intercept the same arc. The angles are not
congruent and the ratio is not 1 if the angles do not intercept the same arc.
TI-Nspire Navigator Opportunity: Screen Capture
See Note 4 at the end of this lesson.
Teacher Tip: Some students may note that the angle measurements are
the same whenever AD and BC intersect or “criss-cross” and are not the
same when they don’t. This observation is important but needs to be
related to intercepted arcs. Some students may notice that the angles are
supplementary when they do not intercept the same arc. This property is
addressed in problem 9. If m∠ABC equals 90°, then m∠ADC equals 90°
whether or not they share the same intercepted arc.
7.
Place point D at the center of the circle. Move point A and point C so that ∠ADC intercepts the same
arc as ∠ABC.
a. What is the relationship between the measures of inscribed ∠ABC and central ∠ADC?
Answer: The measure of the central angle is double the measure of the inscribed angle. The
measure of the inscribed angle is half the measure of the central angle. The ratio of the measure
of ∠ADC to the measure of ∠ABC is 2.
TI-Nspire Navigator Opportunity: Quick Poll
See Note 5 at the end of this lesson.
b. What happens to the angles if you move point A or point C?
Answer: The measure of the inscribed angle is half the measure of the central angle (the ratio of
the measure of ∠ADC to the measure of ∠ABC is 2), as long as both angles intercept the same
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Circles—Angles and Arcs
TEACHER NOTES
MATH NSPIRED: GEOMETRY
arc.
TI-Nspire Navigator Opportunity: Screen Capture and/or Live Presenter
See Note 6 at the end of this lesson.
Teacher Tip: For a proof of the Inscribed Angle Theorem: In the simplest
case, one leg of the inscribed angle is a diameter of the circle so it passes
through the center of the circle. Since that leg is a straight line, the
supplement of the central angle equals 180° − 2θ. Drawing a segment from
the center of the circle to the other point of intersection of the inscribed
angle produces an isosceles triangle, made from two radii of the circle and
the second leg of the inscribed angle. Since two angles in an isosceles
triangle are equal and since the angles in a triangle sum to 180°, it follows
that the inscribed angle equals θ, half of the central angle.
8. Leona said, “Since a central angle can never measure more than 180°, I know an inscribed angle can
never measure more than 90°.” Do you agree or disagree? Why?
Answer: I disagree because the central angle always intercepts a minor arc, but an inscribed angle
can intercept a major arc.
9. Place point D on the circle so that ABCD is a quadrilateral.
a. What do you notice about the sum of the measures of ∠ABC and ∠ADC? Check with a
classmate to compare.
Answer: The sum of the measures of ∠ABC and ∠ADC is 180°.
b. What do you notice about the sum of the measures of the angles if you move point A or point C?
Answer: As long as ABCD remains a quadrilateral, the sum of the measures of ∠ABC and
∠ADC remains 180°.
c.
What do you notice about arcs ABC and ADC?
Answer: One is a major arc and one is a minor arc. Together the arcs make a circle. The
measures of the arcs sum to 360°.
d. How does the relationship between arcs ABC and ADC explain the sum of the measures of
inscribed ∠ABC and ∠ADC?
Answer: The sum of the measures of the major and minor arcs is 360°. Since the measures of
the inscribed angles are half the measures of their intercepted arcs, the angles are
supplementary.
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Circles—Angles and Arcs
TEACHER NOTES
MATH NSPIRED: GEOMETRY
Teacher Tip: A quadrilateral inscribed in a circle has the special name
“cyclic quadrilateral.”
TI-Nspire Navigator Opportunity: Quick Poll
See Note 7 at the end of this lesson.
Wrap Up:
Upon completion of the discussion, the teacher should ensure that students understand:
•
Two inscribed angles intercepting the same arc have the same measure.
•
An inscribed angle measure of 90° results in the endpoints of the intercepted arc lying on a diameter.
•
The measure of the inscribed angle is half the measure of the central angle that intercepts the same
arc.
TI-Nspire Navigator
Note 1
Questions 4a and 4b, Quick Poll: Have students enter their answers to the first
part of questions 4a and 4b in Quick Poll. Ask students to answer, “How Do You
Know?” orally.
Note 2
Question 5c, Screen Capture: As students complete question 5c, use Screen
Capture to view their screens. Discuss the following:
•
Are all the right angles located in exactly the same place?
•
What is true about the arc intercepted by the right angle?
•
What is the measure of the intercepted arc?
•
What is AC on each screen if a right angle is shown?
Note 3
Question 6a, Quick Poll: Using the Open Response feature of Quick Poll, have
students enter their answer to 6a.
Note 4
Question 6b, Screen Capture: Use Screen Capture to look at students’ screens
when they complete question 6b. Put the screens together where the angles
intercept the same arc and the screens together where the angles do not
intercept the same arc. Discuss the results.
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Circles—Angles and Arcs
TEACHER NOTES
MATH NSPIRED: GEOMETRY
Note 5
Question 7a, Quick Poll: Using Quick Poll, students answer the following:
“The ratio of the measure of ∠ADC to the measure of ∠ABC is ____.” Discuss
students’ responses.
Note 6
Question 7b, Screen Capture: Use Screen Capture to discuss students’
answers to question 7b. Screen Capture can also be used to help students
answer the extension questions.
Note 7
Wrap Up, Quick Poll: Use Quick Poll to be sure students understand the three
statements listed under Wrap Up. The True/False feature would be a good
choice.
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Application of a Circle – Angles and Arcs
TEACHER NOTES
MATH NSPIRED: GEOMETRY
Math Objectives
• Students will identify and know the difference between central
angles and inscribed angles of a circle.
• Students will identify the relationships between the measures of
inscribed angles and the arcs they intercept.
• Students will identify and know the difference between major arcs
and minor arcs.
• Students will identify and find the diameter, circumference, and
TI-Nspire™ Technology Skills:
area of a circle given the appropriate formulas.
• Students will see an example of how circles can be used in a real-
• Download a TI-Nspire
document
world application.
• Open a document
Vocabulary
• Move between pages
• central angle
• Grab and drag a point
• inscribed angle
• Find the length of a segment
• major arc
• Find the length of a circle
• minor arc
• Find the area of a circle
• diameter
• Create a segment
• radius
• circumference
Tech Tips:
• area
• Make sure the font size on
your TI-Nspire handhelds is
set to Medium.
About the Lesson
• This lesson is a follow-up lesson to the activity Circles – Angles and
Arcs.
• This lesson involves using relationships among different types of
angles and arcs in a circle to solve a real-world application involving
the design of a courtyard.
• Students will use basic circle concepts, such as inscribed angles, to
manipulate the design to desired specifications.
• Students will use other circle concepts, such as diameter, radius,
circumference, and area, to determine basic information needed to
construct the courtyard.
Lesson Materials:
Student Activity
Application_of_a_Circle
_Angles_and_Arcs
_Student.pdf
Application_of_a_Circle
_Angles_and_Arcs
_Student.doc
TI-Nspire document
Application_of_a_Circle
_Angles_and_Arcs.tns
TI-Nspire™ NavigatorTM System
Visit www.mathnspired.com for
• Use Teacher Edition computer software to review student
lesson updates and tech tip
videos.
documents.
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Application of a Circle – Angles and Arcs
TEACHER NOTES
MATH NSPIRED: GEOMETRY
Discussion Points and Possible Answers
Tech Tip: If students experience difficulty dragging a point, check to make
sure that they have moved the cursor until it becomes a hand (÷) getting
ready to grab the point. Also, be sure that the word point appears, not the
word text. Then press /
x to grab the point and close the hand ({).
Move to page 1.2.
Suppose you work for an architectural firm and a new business complex is in the process of
being designed. The plans for the complex include a circular courtyard within a square area
with side lengths of 8.4 yards. The courtyard will use 10-inch square pavers in different
colors to create a design, as shown in the diagram on page 1.4. The points A, B, C, D, and
E represent the points of the star design, and each of the points lies on the circle. Point Q
represents the center of the circle.
You have been asked to supply the company constructing the courtyard some information
that will help with creating the design and ordering supplies.
Move to page 1.4.
1. Would the angles A, B, C, D, and E be considered central angles
or inscribed angles? Explain.
Answer: The angles would be inscribed angles since the vertices are on the circle and the sides of
the angles are chords of circle Q. The vertex of a central angle is at the center of the circle.
2. What is the relationship between the measures of the angles A, B, C, D, and E and the arcs they
intercept? Explain.
Answer: A central angle has the same degree measure as the arc it intercepts. An inscribed angle
is half the degree measure of the central angle. Therefore, the degree measures of the inscribed
angles of circle Q are half the measures of the arcs they intercept.
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Application of a Circle – Angles and Arcs
TEACHER NOTES
MATH NSPIRED: GEOMETRY
Teacher Tip: Students may need help coming to this conclusion. Providing
a hint such as using the relationship between the measures of central
angles and the arcs they intercept may be helpful.
3. Would the arcs intercepted by each of the angles A, B, C, D, and E be considered major arcs or
minor arcs? Explain.
Answer: A major arc is a part of a circle that measures between 180 and 360 degrees. A minor arc
measures less than 180 degrees. Therefore, the arcs intercepted by each of the angles would be
minor arcs.
4. In order for the star pattern to be uniform, each of the angles should have the same degree
measure. What should be the degree measure of each of the angles A, B, C, D, and E? Explain
your reasoning.
Answer: Since 360 degrees represents the circle and there are 5 angles, divide 360 by 5 to get 72
degrees. An inscribed angle is half the measure of the intercepted arc. Divide 72 by 2 to get 36
degrees. Each of the angles should measure 36 degrees.
5. Grab and move points A, B, C, D, and E around the circle until all
angle measures are the same. What is the degree measure? Is
this what you expected? Explain.
Sample Answer: The degree measure of each of the five angles
is 36 degrees. This result will correspond with the answer to
Question 4, if students answered correctly.
6. What is the diameter of circle Q? Explain your reasoning.
Answer: By looking at the diagram and visualizing a diameter of circle Q that is parallel with two
sides of the square, it appears as though the diameter will be the same as the length of the sides of
the square in which it is inscribed.
Check your answer to Question 6 by using the Segment tool (MENU > Points & Lines > Segment) to
create a segment that represents the diameter of circle Q. Use the Length tool (MENU > Measurement
> Length) to find the length of the segment. Change the Attributes (MENU > Actions > Attributes) of
the measurement to one decimal place.
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Application of a Circle – Angles and Arcs
TEACHER NOTES
MATH NSPIRED: GEOMETRY
7. Given that the circumference of a circle can be found by using the formula C = 2 π r, where r is the
radius, find the circumference of circle Q to the nearest yard. Show your work below.
Answer: C = 2 π r = 2 · π · 4.2 = 26.389 = 26 yd
Check your answer to Question 7 by using the Length tool to find the length of circle Q. Change the
Attributes of the measurement to display zero decimals.
8. Given that the area of a circle can be found by using the formula A = π r2, where r is the radius, find
the area of circle Q to the nearest yard. Show your work below.
Answer: A = π r2 = π · 4.22 = 55.418 = 55 yd2
Check your answer to Question 8 by using the Area tool (MENU > Measurement > Area) to find the
area of circle Q. Change the Attributes of the measurement to display zero decimals.
9. Given that the pavers being used to construct the courtyard are squares with side lengths of 10
inches, how many pavers will need to be ordered to construct the courtyard? (1 yd2 = 1,296 in.2)
Answer: First, convert the area of the circle to inches, since the pavers are given in units of inches,
and then divide by 100 since the pavers are 10-inch squares. At least 713 pavers would need to be
ordered to construct the courtyard.
1,296 in.2
⋅ 55 yd2 = 71,280 in.2 ÷ 100 in.2 = 712.8
1 yd2
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Application of a Circle – Angles and Arcs
TEACHER NOTES
MATH NSPIRED: GEOMETRY
Wrap Up
Upon completion of the discussion, the teacher should ensure that students understand:
•
The difference between central angles and inscribed angles of a circle.
•
The relationship between the measure of inscribed angles and the arcs they intercept.
•
The difference between a major arc and a minor arc.
•
How to identify the diameter of a circle.
•
The relationship between the diameter and radius of a circle.
•
How to find the circumference and area of a circle.
Assessment
Complete a similar problem by changing the dimensions of the square in which the circle is inscribed, or
have students design their own courtyard using both inscribed and central angles.
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