central angles, inscribed angles, and intercepted arcs...the inscribed angle theorem states: “the...

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LEaRninggoaLS KEyTERMS

• degree measure of an arc• adjacent arcs• Arc Addition Postulate• intercepted arc• Inscribed Angle Theorem• Parallel Lines–Congruent Arcs Theorem

In this lesson, you will:

• Determine the measures of arcs .• Use the Arc Addition Postulate .• Determine the measures of central angles

and inscribed angles .• Prove the Inscribed Angle Theorem .• Prove the Parallel Lines–Congruent

Arcs Theorem .

TaketheWheelCentralangles,inscribedangles,andinterceptedarcs

Before airbags were installed in car steering wheels, the recommended position for holding the steering wheel was the 10–2 position. Now, one of the recommended

positions is the 9–3 position to account for the airbags. The numbers 10, 2, 9, and 3 refer to the numbers on a clock. So, the 10–2 position means that one hand is at 10 o’clock and the other hand is at 2 o’clock.

9.2

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Problem 1 KeepBothHandsontheWheel

Recall that the degree measure of a circle is 360° .

Each minor arc of a circle is associated with and determined by a specific central angle . The degree measure of a minor arc is the same as the degree measure of its central angle . For example, if the measure of central angle PRQ is 30°, then the degree measure of its minor arc PQ is equal to 30° . Using symbols, this can be expressed as follows: If PRQ is a central angle and mPRQ 5 30°, then m  ⁀ PQ  5 30° .

1. The circles shown represent steering wheels, and the points on the circles represent the positions of a person’s hands .

A

O

B

C D

P

For each circle, use the given points to draw a central angle . The hand position on the left is 10–2 and the hand position on the right is 11–1 .

a. What are the names of the central angles?

b. Without using a protractor, determine the central angle measures . Explain your reasoning .

c. How do the measures of these angles compare?

What if angle CPD is 30˚? What would the

measure of ⁀  CD be?

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9.2 Central Angles, Inscribed Angles, and Intercepted Arcs 663

d. Why do you think the hand position represented by the circle on the left is recommended and the hand position represented on the right is not recommended?

e. Describe the measures of the minor arcs .

f. Plot and label point Z on each circle so that it does not lie between the endpoints of the minor arcs . Determine the measures of the major arcs that have the same endpoints as the minor arcs .

2. If the measures of two central angles of the same circle (or congruent circles) are equal, are their corresponding minor arcs congruent? Explain your reasoning .

3. If the measures of two minor arcs of the same circle (or congruent circles) are equal, are their corresponding central angles congruent? Explain your reasoning .

What is the measure of

a semicircle?

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Adjacent arcs are two arcs of the same circle sharing a common endpoint .

4. Draw and label two adjacent arcs on circle O shown .

O

The Arc Addition Postulate states: “The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs .”

5. Apply the Arc Addition Postulate to the adjacent arcs you created .

An intercepted arc is an arc associated with and determined by angles of the circle . An intercepted arc is a portion of the circumference of the circle located on the interior of the angle whose endpoints lie on the sides of an angle .

6. Consider circle O .

a. Draw inscribed PSR on circle O .

b. Name the intercepted arc associated with PSR .

7. Consider the central angle shown .

A

B

O

a. Use a straightedge to draw an inscribed angle that contains points A and B on its sides . Name the vertex of your angle point P . What do the angles have in common?

O

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?

b. Use your protractor to measure the central angle and the inscribed angle . How is the measure of the inscribed angle related to the measure of the central angle and the measure of ⁀ AB ?

c. Use a straightedge to draw a different inscribed angle that contains points A and B on its sides . Name its vertex point Q . Measure the inscribed angle . How is the measure of the inscribed angle related to the measure of the central angle and the measure of ⁀ AB ?

d. Use a straightedge to draw one more inscribed angle that contains points A and B on its sides . Name its vertex point R . Measure the inscribed angle . How is the measure of the inscribed angle related to the measure of the central angle and the measure of ⁀ AB ?

8. What can you conclude about inscribed angles that have the same intercepted arc?

9. Dalia says that the measure of an inscribed angle is half the measure of the central angle that intercepts the same arc . Nate says that it is twice the measure . Sandy says that the inscribed angle is the same measure . Who is correct? Explain your reasoning .

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10. Inscribed angles formed by two chords can be drawn three different ways with respect to the center of the circle .

Case 1: Use circle O shown to draw and label inscribed /MPT such that the center point lies on one side of the inscribed angle .

O

Case 2: Use circle O shown to draw and label inscribed /MPT such that the center point lies on the interior of the inscribed angle .

O

Case 3: Use circle O shown to draw and label inscribed /MPT such that the center point lies on the exterior of the inscribed angle .

ODo these

three cases cover all the possible ways inscribed angles can

be drawn?

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11. To prove your Inscribed Angle Conjecture, you must prove each case in Question 10 .

Case 1:

OPM

T

Given: MPT is inscribed in circle O .

mMPT 5 x

Point O lies on diameter ____

PM  .

Prove: m/MPT 5   1 __ 2   m  ⁀ MT 

Statements Reasons

1. /MPT is inscribed in circle O . m/MPT 5 x

Point O lies on diameter ____

PM   .

1. Given

2. Connect points O and T to form radius

___ OT   .

2. Construction

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Case 2:

O

P

M

R

T

Given: MPT is inscribed in circle O .

Point O is in the interior of MPT .

mMPO 5 x

mTPO 5 y

Prove: mMPT 5 1 __ 2 m  ⁀ MT 

Statements Reasons

1. /MPT is inscribed in circle O . Point O is in the interior of /MPT .

m/MPO 5 x m/TPO 5 y

1. Given

2. Construct diameter ___

PR   . Connect points O and T to form radius

___ OT   .

Connect points O and M to form radius

____ OM   .

2. Construction

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Case3:

OP

T

M

R

Given: MPT is inscribed in circle O.

Point O is in the exterior of MPT.

Prove: mMPT 5 1 __ 2 m⁀ MT

Statements Reasons

1. /MPT is inscribed in circle O. Point O is in the exterior of /MPT.

1. Given

2. Construct diameter ___

PR. Connect points O and T to form radius

___ OT.

Connect points O and M to form radius

____ OM.

2. Construction

The InscribedAngleTheorem states: “The measure of an inscribed angle is half the measure of its intercepted arc.”

You will need to construct a diameter

through point P and construct radii OM and OT.

You’ve just proved the Inscribed

Angle Conjecture, which you can now call the Inscribed Angle

Theorem. Nice effort!

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12. Aubrey wants to take a family picture. Her camera has a 70º field of view, but to include the entire family in the picture, she needs to cover a 140º arc. Explain what Aubrey needs to do to fit the entire family in the picture. Use the diagram to draw the solution.

70°

Family in photo

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Problem 2 ParallelLinesintersectingaCircle

Do parallel lines intersecting a circle intercept congruent arcs on the circle?

A

O R

P

L

1. Create a proof for this conjecture .

Given:

Prove:

You have just proven the Parallel Lines–Congruent Arcs Conjecture . It is now known as the Parallel Lines–Congruent Arcs Theorem which states that parallel lines intercept congruent arcs on a circle .

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TalktheTalk

1. ____

MP   is a diameter of circle O .

OP

WT

M

KS

If m  ⁀ MT  5 124° , determine mTPW . Explain your reasoning .

2. Use the diagram shown to answer each question .

O

KJ

F G

a. Are radii ___

OJ  , ___

OK  , ___

OF  , and ____

OG   all congruent? Explain your reasoning .

b. Is m  ⁀ FG  greater than m  ⁀ JK ? Explain your reasoning .

c. If mFOG 5 57°, determine m  ⁀ JK  and m  ⁀ FG  . Explain your reasoning .

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3. DeJaun told Thomas there was not enough information to determine whether circle A was congruent to circle B . He said they would have to know the length of a radius in each circle to determine whether the circles were congruent . Thomas explained to DeJaun why he was incorrect . What did Thomas say to DeJaun?

Be prepared to share your solutions and methods .

A

B

central angles, inscribed angles, and intercepted arcs...the inscribed angle theorem states: “the...

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