GPS Lesson Plan - Paul Markley
Angle Relationships with Circles
Materials:
protractor
Question
What are the relationships between angles and circles?
Launch
For the circle shown, use a straight edge to draw a central angle making sure the sides of
the angle intersect on the tick marks. Measure your angle.
Angle measure: ________
How far apart (in degrees) are the tick marks?
How many tick marks are on the circle?
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Investigation:
The tick marks below are in 10o increments. Draw an angle that has its vertex on
the circle and has sides that intersect the circle through tick marks. This angle is
said to be inscribed. Use your protractor to measure the angle.
Measure of your angle: _________
Measure of the arc which it intercepts: ________
Compare your results with others in your group.
What do you notice?
What would be the formula that would generalize your findings?
Write a conditional statement of this formula. Is its converse true?
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Draw two chords that are not diameters but intersect inside the circle (make sure the
endpoints fall on tick marks). Use your protractor to measure one of the angles. What is
the measure of the intercepted arc? Look at the vertical angle. What is the measure of
that intercepted arc?
Measure of your angle: _________
Measure of the arc which it intercepts: ________
Measure of the arc which vertical angle intercepts: ________
Compare your results with others in your group.
What do you think is the connection between the angle measure and the two intercepted arcs?
What would be the formula that would generalize your findings?
Write a biconditional statement of this formula.
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Draw two secant segments that intersect outside the circle (make each secant
segment intersects two tick marks). Use your protractor to measure one of the
angles. What is the measure of the larger intercepted arc? What is the measure
of the smaller intercepted arc?
Measure of your angle: _________
Measure of the larger arc which it intercepts: ________
Measure of the smaller arc which it intercepts: ________
Compare your results with others in your group.
What do you think is the connection between the angle measure and the two intercepted arcs?
(hint: it is much like the previous circle but with a different operation)
What would be the formula that would generalize your findings?
Write a biconditional statement of this formula.
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Draw, as best as you can, a tangent to the circle. Draw a chord from the point of tangency to a
tick mark. Use your protractor to measure the acute angle. What is the measure of its
intercepted arc?
Measure of your angle: _________
Measure of the arc which it intercepts: ________
Compare your results with others in your group.
What do you think is the connection between the angle measure and the intercepted arc?
Also, does the vertex fall inside, outside, or on the circle?
What would be the formula that would generalize your findings?
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Conclusions (In-Class Problems)
1. If an angle that measures xo is inscribed in a circle, what is the measure of the
arc that it intercepts?
2. The points A, B, C, and D are on a circle and AB intersects CD at P. If
then P is ? (inside, outside, or on) the circle.
3. If a circle is circumscribed about a polygon, then the polygon is ? in the circle.
4. In the diagram to the right, what is the
measure of angle B?
A
mAC = 116
D
What is the measure of major arc ABC?
C
What is the measure of angle D?
B
Add the measures of angle B and angle D.
What do you notice?
Make a conjecture about the angles of a quadrilateral that are
inscribed in a circle.
5. In the diagram to the right, what is the
measure of angle B?
C
C
What can you say about inscribed angles
that intercept a semicircle?
B
Using this knowledge, describe a way to find the center of any circle.
Closure
The relationships between angles and circles depend on where the vertex is. If the vertex
is on the circle, m  ________. If the vertex is inside the circle, m  _____________. If the
vertex is outside the circle, m  _____________.
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Exercises
1. If the chord of a circle intersects a tangent to the circle at the point of tangency, what is
the relationship formed between the angles formed and the intercepted arcs?
2. Can the parallelogram to the right be inscribed in a circle?
Why or why not?
Find the indicated measure.
9. Name two pair of congruent angles.
Find the value of x.
10.
11.
110
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12.
13.
Line t is tangent to the circle. Find the indicated measure.
14.
16.
15.
17. Find x.
m1
18. In the diagram at the right, PL is tangent to the
circle and KJ is a diameter. What is the range
of possible angle measures of LPJ ? Explain.
19. Find the values of x and y:
78
30
x
y
170
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Standards Addressed
(Though this is a Mathematics II unit, Mathematics I standards are included)
MM1G2. Students will understand and use the language of mathematical argument and
justification.
a. Use conjecture, inductive reasoning, deductive reasoning, counterexamples,
and indirect proof as appropriate.
b. Understand and use the relationships among a statement and its converse,
inverse, and contrapositive.
MM1P1. Students will solve problems (using appropriate technology).
d. Monitor and reflect on the process of mathematical problem solving.
MM2G3. Students will understand the properties of circles.
a. Understand and use properties of chords, tangents, and secants as an
application of triangle similarity.
b. Understand and use properties of central, inscribed, and related angles.
d. Justify measurements and relationships in circles using geometric and
algebraic properties.
MM2P2. Students will reason and evaluate mathematical arguments.
b. Make and investigate mathematical conjecture.
MM2P3. Students will communicate mathematically.
a. Organize and consolidate their mathematical thinking through communication.
b. Communicate their mathematical thinking coherently and clearly to peers, teachers,
and others.
c. Analyze and evaluate the mathematical thinking and strategies of others.
d. Use the language of mathematics to express mathematical ideas precisely.
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Teacher Notes:
before the lesson, students should be familiar with the
following: central angle, measure of an arc, biconditional
statements/definitions, chord, tangent, tangent segment, secant,
and secant segment.
In the discovery part of this lesson, students are asked to
compare their results. If this is not feasible, draw a table on the
board and compare several classmates’ results or have the
students do more than one investigation for each circle.
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