CorrectionKey=NL-C;CA-C Name Class Date 15 . 5 Angle ... · 2 Explain 1 Proving the Intersecting...

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Explore Exploring Angle Measures in CirclesThe sundial is one of many instruments that use angles created in circles for practical applications, such as telling time.

In this lesson, you will observe the relationships between angles created by various line segments and their intercepted arcs.

Using geometry software, construct a circle with two secants

‹−›CD and ‹−›EF that intersect inside the circle at G,

as shown in the figure.

Create two new points H and I that are on the circle as shown. These will be used to measure the arcs. Hide B if desired.

Measure ∠DGF formed by the secant lines, and measure ⁀CHE and ⁀DIF . Record angle and arc measurements in the first column of the table.

m∠DGF

m ⁀ CHE

m ⁀ DIF

Sum of Arc Measures

Drag F around the circle and record the changes in measures in the table in Part C. Try to create acute, right, and obtuse angles. Be sure to keep H between C and E and I between D and F for accurate arc measurement. Move them if necessary.

Module 15 829 Lesson 5

15 . 5 Angle Relationships in CirclesEssential Question: What are the relationships between angles formed by

lines that intersect a circle?

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Reflect

1. Can you make a conjecture about the relationship between the angle measure and the two arc measures?

2. Using the same circle you created in step A, drag points around the circle so that the intersection is outside the circle, as shown. Measure ∠FGC formed by the secant lines and measure ⁀ CIF and ⁀ DHE . Drag points around the circle and observe the changes in measures. Record some measures in the table.

What is similar and different about the relationships between the angle measure and the arc measures when the secants intersect outside the circle?

2 Explain 1 Proving the Intersecting Chords Angle Measure Theorem

In the Explore section, you discovered the effects that line segments, such as chords and secants, have on angle measures and their intercepted arcs. These relationships can be stated as theorems, with the first one about chords.

The Intersecting Chords Angle Measure Theorem

If two secants or chords intersect in the interior of a circle, then the measure of each angle formed is half the sum of the measures of its intercepted arcs.

Chords _ AD and

_ BC intersect at E.

m∠1 = 1 _ 2 (m ⁀ AB + m ⁀ CD )

m∠FGC

m ⁀ CIF

m ⁀ DHE

Difference of Arc Measures



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Example 1 Prove the Intersecting Chords Angle Measure Theorem

Given: ̄ AD and ̄ BC intersect at E.

Prove: m∠1 = 1 _ 2 (m ⁀ AB + m ⁀ CD )

Statements Reasons

1. _ AD and

_ BC intersect at E. 1. Given

2. Draw _ BD . 2. Through any two points, there is exactly

one line.

3. m∠1 = m∠EDB + m∠EBD 3.

4. m∠EDB = 1 __ 2 m ⁀ AB ,

m∠EBD = 1 __ 2 m ⁀ CD

4.

5. m∠1 = 1 __ 2 m ⁀ AB + 1 __ 2 m 5. Substitution Property

6. 6.

Reflect

3. DIscusssion Explain how an auxiliary segment and the Exterior Angle Theorem are used in the proof of the Intersecting Chords Angle Measure Theorem.

Your Turn

Find each unknown measure.

4. m∠MPK 5. m ⁀ PR



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Explain 2 Applying the Tangent-Secant Interior Angle Measure Theorem

The angle and arc formed by a tangent and secant intersecting on a circle also have a special relationship.

The Tangent-Secant Interior Angle Measure Theorem

If a tangent and a secant (or a chord) intersect on a circle at the point of tangency, then the measure of the angle formed is half the measure of its intercepted arc.

Tangent ‾→ BC and secant

‾→ BA intersect at B.

m∠ABC = 1 _ 2 m ⁀ AB

Example 2 Find each unknown measure.

m∠BCD

m∠BCD = 1 _ 2

m ⁀ BC

= 1 _ 2

(142°)

= 71°

m ⁀ ABC

m∠ACD = 1 _ 2

(m ⁀ ABC ) = 1 _

2 (m ⁀ ABC )

= m ⁀ ABC

Your Turn

Find the measure.

6. m ⁀ PN 7. m∠MNP

Explain 3 Applying the Tangent-Secant Exterior Angle Measure Theorem

You can use the difference in arc measures to find measures of angles formed by tangents and secants intersecting outside a circle.



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83°25°

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The Tangent-Secant Exterior Angle Measure Theorem

If a tangent and a secant, two tangents, or two secants intersect in the exterior of a circle, then the measure of the angle formed is half the difference of the measures of its intercepted arcs.

m∠1 = 1 _ 2 (m ⁀ AD - m ⁀ BD ) m∠2 = 1 _ 2 (m ⁀ EHG - m ⁀ EG ) m∠3 = 1 _ 2 (m ⁀ JN - m ⁀ KM )

Example 3 Find the value of x.

Your Turn

Find the value of x.

8.

x° = 1 _ 2 (238° - (360° - ) ) x = 1 _ 2 ( (238 - ) ) x = 1 _ 2 ( ) x =

m∠L = 1 _ 2

(m ⁀ JN - m ⁀ KM ) 25° = 1 _

2 (83° - x°)

50 = 83 - x

-33 = - x

33 = x



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1 78°202° 2 45° 125°

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9. The superior oblique and inferior oblique are two muscles that help control eye movement. They intersect behind the eye to create an angle, as shown. If m ⁀ AEB = 225°, what is m∠ACB?

Explain 4 Understanding Angle Relationships in CirclesYou can summarize angle relationships in circles by looking at where the vertex of the angle lies: on the circle, inside the circle, or outside the circle.

Angle Relationships in Circles

Vertex of the Angle Measure of Angle Diagrams

On a circle Half the measure of its intercepted arc

m∠1 = 60° m∠2 = 100°Inside a circle Half the sum of

the measures of its intercepted arcs

m∠1 = 1 _ 2

(44° + 86°) = 65°

Outside a circle Half the difference of the measures of its intercepted arcs

m∠1 = 1 _ 2

(202° - 78°) m∠2 = 1 _ 2

(125° - 45°) = 62° = 40°


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Angle’s VertexLocation

Inside theCircle

Outside theCircle

The angle measure ishalf the measure ofits intercepted arc.

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Example 4 Find the unknown arc measures.

Find m ⁀ FD .

m∠E = 1 _ 2

(m ⁀ BF - m ⁀ FD ) 50° = 1 _

2 (150° - m ⁀ FD )

100° = (50° - m ⁀ FD ) -50° = -m ⁀ FD

50° = m ⁀ FD

Find m ⁀ CD .

m ⁀ CD = - (m ⁀ BC + m ⁀ BF + mF ⁀ D ) = - (64° + + ) = -

=

Your Turn

10. Find m ⁀ KN .

Elaborate

11. Complete the graphic organizer that shows the relationship between the angle measurementand the location of its vertex.

12. Essential Question Check-In What is similar about all the relationships between angle measures and their intercepted arcs?



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Use the figure for Exercises 1−2.

Suppose you use geometry software to construct a secant ‹ −

› CE and tangent ‹ −

› CD that intersect on a circle at point C.

1. Suppose you measure ∠DCE and you measure ⁀ CB E. Then you drag the points around the circle and measure the angle and arc three more times. What would you expect to find each time? Which theorem from the lesson would you be demonstrating?

2. When the measure of the intercepted arc is 180°, what is the measure of the angle? What does that tell you about the secant?

Find each measure.

3. m∠QPR 4. m∠ABC

5. m∠MKJ 6. m∠NPK

• Online Homework• Hints and Help• Extra Practice

Evaluate: Homework and Practice



112°

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Find each measure. Use the figure for Exercises 7–8.

7. m∠BCD

8. m∠ABC

Find each measure. Use the figure for Exercises 9–10.

9. m∠XZW 10. m∠YXZ

Find the value of x.

11. 12.

13.



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137° 89°

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14. Represent Real-World Problems Stonehenge is a circular arrangement of massive stones near Salisbury, England. A viewer at V observes the monument from a point where two of the stones A and B are aligned with stones at the endpoints of a diameter of the circular shape. Given that m ⁀ AB = 48°, what is m∠AVB?

15. Multi-Step Find each measure.

a. Find m ⁀ PN .

b. Use your answer to part a to find m ⁀ KN .

16. Multi-Step Find each measure.

a. Find m ⁀ DE .

b. Use your answer to part a to find m∠F.



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‹ −

› MS || ‹ − › PQ and m∠PNS = 50°. Find each measure.

17. m ⁀ PR

18. m ⁀ LP

19. Represent Real-World Problems A satellite orbits Mars. When it reaches S it is about 12,000 km above the planet. What is x°, the measure of the arc that is visible to a camera in the satellite?

20. Use the circle with center J. Match each angle or arc on the left with its measure on the right. Indicate a match by writing the letter for the angle or arc on the line in front of the corresponding measure.

A. ∠BAE 41°

B. ∠ACD 180°

C. ⁀ AF 101°

D. ∠AED 90°

E. ⁀ ADE 60°



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21. Use the Plan for Proof to write a proof for one case of the Tangent-Secant Exterior Angle Measure Theorem.

Given: Tangent ‾→ CD and secant

‾→ CA

Prove: m∠ACD = 1 _ 2 (m ⁀ AD - m ⁀ BD ) Plan: Draw auxiliary line segment

_ BD . Use the Exterior Angle Theorem to show that

m∠ACD = m∠ABD - m∠BDC. Then use the Inscribed Angle Theorem and the Tangent-Secant Interior Angle Measure Theorem.

22. Justify Reasoning Write a proof that the figure shown is a square. Given:

_ YZ and

_ WZ are tangent to circle X, m ⁀ WY = 90°

Prove: WXYZ is a square.



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H.O.T. Focus on Higher Order Thinking

23. Justify Reasoning Prove the Tangent-Secant Interior Angle Theorem. Given: Tangent

‾→ BC and secant

‾→ BA

Prove: m∠ABC = 1 _ 2 m ⁀ AB

(Hint: Consider two cases, one where _ AB is a diameter and one where

_ AB is not a diameter.)

24. Critical Thinking Suppose two secants intersect in the exterior of a circle as shown. Which is greater, m∠1 or m∠2? Justify your answer.



A

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EB D 0.5°

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The diameter of the Moon is about 2160 miles. From Earth, the portion of the Moon’s surface that an observer can see is from a circumscribed angle of approximately 0.5°.

a. Find the measure of ⁀ ADC . Explain how you found the measure.

b. What fraction of the circumference of the Moon is represented by ⁀ ADC ?

c. Find the length of ⁀ ADC . You can use the formula C = 2πr to find the circumference of the Moon.

Lesson Performance Task



CorrectionKey=NL-C;CA-C Name Class Date 15 . 5 Angle ... · 2 Explain 1 Proving the Intersecting...

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