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Inscribed Angles in Circles
Bill Zahner
Lori Jordan
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Printed: January 24, 2016
AUTHORS
Bill Zahner
Lori Jordan
www.ck12.org
C HAPTER
Chapter 1. Inscribed Angles in Circles
1
Inscribed Angles in Circles
Here you’ll learn the properties of inscribed angles and how to apply them.
What if your family went to Washington DC over the summer and saw the White House? The closest you can get to
the White House are the walking trails on the far right. You got as close as you could (on the trail) to the fence to
take a picture (you were not allowed to walk on the grass). Where else could you have taken your picture from to get
the same frame of the White House? Where do you think the best place to stand would be? Your line of sight in the
camera is marked in the picture as the grey lines. The white dotted arcs do not actually exist, but were added to help
with this problem. After completing this Concept, you will be able to use inscribed angles to answer this question.
Watch This
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CK-12 Foundation: Chapter9InscribedAnglesinCirclesA
Learn more about inscribed angles by watching the video at this link.
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Guidance
An inscribed angle is an angle with its vertex is the circle and its sides contain chords. The intercepted arc is the
arc that is on the interior of the inscribed angle and whose endpoints are on the angle. The vertex of an inscribed
angle can be anywhere on the circle as long as its sides intersect the circle to form an intercepted arc.
Let’s investigate the relationship between the inscribed angle, the central angle and the arc they intercept.
Investigation: Measuring an Inscribed Angle
Tools Needed: pencil, paper, compass, ruler, protractor
J
1. Draw three circles with three different inscribed angles. For A, make one side of the inscribed angle a diameter,
J
J
for B, make B inside the angle and for C make C outside the angle. Try to make all the angles different sizes.
2. Using your ruler, draw in the corresponding central angle for each angle and label each set of endpoints.
3. Using your protractor measure the six angles and determine if there is a relationship between the central angle,
the inscribed angle, and the intercepted arc.
m6 LAM =
m6 NBP =
c =
mLM
c=
mNP
m6 QCR =
c=
mQR
m6 LKM =
m6 NOP =
m6 QSR =
Inscribed Angle Theorem: The measure of an inscribed angle is half the measure of its intercepted arc.
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Chapter 1. Inscribed Angles in Circles
c If we had drawn in the central angle 6 ABC, we could also say that m6 ADC =
In the picture, m6 ADC = 12 mAC.
1 6
2 m ABC because the measure of the central angle is equal to the measure of the intercepted arc. To prove the
Inscribed Angle Theorem, you would need to split it up into three cases, like the three different angles drawn from
the Investigation.
Congruent Inscribed Angle Theorem: Inscribed angles that intercept the same arc are congruent.
Inscribed Angle Semicircle Theorem: An angle that intercepts a semicircle is a right angle.
In the Inscribed Angle Semicircle Theorem we could also say that the angle is inscribed in a semicircle. Anytime a
right angle is inscribed in a circle, the endpoints of the angle are the endpoints of a diameter. Therefore, the converse
of the Inscribed Angle Semicircle Theorem is also true.
Example A
c and m6 ADB.
Find mDC
c = 2 · 45◦ = 90◦ . m6 ADB = 1 · 76◦ = 38◦ .
From the Inscribed Angle Theorem, mDC
2
Example B
Find m6 ADB and m6 ACB.
c Therefore, m6 ADB = m6 ACB = 1 · 124◦ = 62◦
The intercepted arc for both angles is AB.
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Example C
Find m6 DAB in
J
C.
Because C is the center, DB is a diameter. Therefore, 6 DAB inscribes semicircle, or 180◦ . m6 DAB = 12 · 180◦ = 90◦ .
Watch this video for help with the Examples above.
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CK-12 Foundation: Chapter9InscribedAnglesinCirclesB
Concept Problem Revisited
You can take the picture from anywhere on the semicircular walking path. The best place to take the picture is
subjective, but most would think the pale green frame, straight-on, would be the best view.
Guided Practice
c m6 MNP, m6 LNP, and mLN.
c
Find m6 PMN, mPN,
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Chapter 1. Inscribed Angles in Circles
Answers:
m6 PMN = m6 PLN = 68◦ by the Congruent Inscribed Angle Theorem.
c = 2 · 68◦ = 136◦ from the Inscribed Angle Theorem.
mPN
m6 MNP = 90◦ by the Inscribed Angle Semicircle Theorem.
m6 LNP = 12 · 92◦ = 46◦ from the Inscribed Angle Theorem.
c we need to find m6 LPN. 6 LPN is the third angle in 4LPN, so 68◦ + 46◦ + m6 LPN = 180◦ . m6 LPN =
To find mLN,
◦
c = 2 · 66◦ = 132◦ .
66 , which means that mLN
Interactive Practice
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Explore More
Fill in the blanks.
1.
2.
3.
4.
An angle inscribed in a ________________ is 90◦ .
Two inscribed angles that intercept the same arc are _______________.
The sides of an inscribed angle are ___________________.
J
Draw inscribed angle 6 JKL in M. Then draw central angle 6 JML. How do the two angles relate?
Find the value of x and/or y in
J
A.
5.
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6.
7.
8.
9.
Solve for x.
10.
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Chapter 1. Inscribed Angles in Circles
11.
12.
13.
14. Suppose that AB is a diameter of a circle centered at O, and C is any other point on the circle. Draw the line
c Explain why D is the midpoint of
through O that is parallel to AC, and let D be the point where it meets BC.
c
BC.
15. Fill in the blanks of the Inscribed Angle Theorem proof.
Given: Inscribed 6 ABC and diameter BD
c
Prove: m6 ABC = 12 mAC
TABLE 1.1:
Statement
1. Inscribed 6 ABC and diameter BD
m6 ABE = x◦ and m6 CBE = y◦
2. x◦ + y◦ = m6 ABC
3.
4.
5. m6 EAB = x◦ and m6 ECB = y◦
6. m6 AED = 2x◦ and m6 CED = 2y◦
Reason
All radii are congruent
Definition of an isosceles triangle
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TABLE 1.1: (continued)
Statement
c = 2x◦ and mDC
c = 2y◦
7. mAD
8.
c = 2x◦ + 2y◦
9. mAC
10.
c = 2m6 ABC
11. mAC
c
12. m6 ABC = 1 mAC
Reason
Arc Addition Postulate
Distributive PoE
2
Answers for Explore More Problems
To view the Explore More answers, open this PDF file and look for section 9.5.
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