Eureka Math™ Exit Ticket Packet Geometry Module 5

$Eureka Math™ Exit Ticket Packet Geometry Module 5$

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Eureka Math™ Geometry Exit Ticket Packet Module 5

Topic A Topic C Lesson 1 Exit Ticket Qty: 30 Lesson 11 Exit Ticket Qty: 30 Lesson 2 Exit Ticket Qty: 30 Lesson 12 Exit Ticket Qty: 30 Lesson 3 Exit Ticket Qty: 30 Lesson 13 Exit Ticket Qty: 30 Lesson 4 Exit Ticket Qty: 30 Lesson 14 Exit Ticket Qty: 30 Lesson 5 Exit Ticket Qty: 30 Lesson 15 Exit Ticket Qty: 30 Lesson 6 Exit Ticket Qty: 30 Lesson 16 Exit Ticket Qty: 30

Topic B Topic D Lesson 7 Exit Ticket Qty: 30 Lesson 17 Exit Ticket Qty: 30 Lesson 8 Exit Ticket Qty: 30 Lesson 18 Exit Ticket Qty: 30 Lesson 9 Exit Ticket Qty: 30 Lesson 19 Exit Ticket Qty: 30 Lesson 10 Exit Ticket Qty: 30 Topic E Lesson 20 Exit Ticket Qty: 30 Lesson 21 Exit Ticket Qty: 30

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^dKZzK&&hEd/KE^ Lesson 1 M5 GEOMETRY

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Lesson 1: Thales’ Theorem

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Circle ܣ is shown below. 1. Draw two diameters of the circle. 2. Identify the shape defined by the endpoints of the two diameters. 3. Explain why this shape is always the result.

Lesson 1: Thales’ Theorem 1

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Lesson 2: Circles, Chords, Diameters, and Their Relationships

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1. Given circle ܣ shown, ܣܨ ൌ ܣܩ and ܤܥ ൌ ʹʹ. Find ܦܧ.

.തܧതതതܦ٣ തതതܤܣIn the figure, circle ܲ has a radius of ͳͲ. ത .2 a. If ܣܤ ൌ ͺ, what is the length of ܣܥതതതത?

b. If ܦܥ ൌ ʹ, what is the length of തܣܤതതത?

Lesson 2: Circles, Chords, Diameters, and Their Relationships 1

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Lesson 3: Rectangles Inscribed in Circles

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Rectangle ܣܤܥܦ is inscribed in circle ܲ. Boris says that diagonal ܣܥതതതത could pass through the center, but it does not have to pass through the center. Is Boris correct? Explain your answer in words, or draw a picture to help you explain your thinking.

Lesson 3: Rectangles Inscribed in Circles 1

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Lesson 4: Experiments with Inscribed Angles

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Joey marks two points on a piece of paper, as we did in the Exploratory Challenge, and labels them ܣ and ܤ. Using the trapezoid shown below, he pushes the acute angle through points ܣ and ܤ from below several times so that the sides of the angle touch points ܣ and ܤ, marking the location of the vertex each time. Joey claims that the shape he forms by doing this is the minor arc of a circle and that he can form the major arc by pushing the obtuse angle through points ܣ and ܤ from above. “The obtuse angle has the greater measure, so it will form the greater arc,” states Joey. Ebony disagrees, saying that Joey has it backwards. “The acute angle will trace the major arc,” claims Ebony.

1. Who is correct, Joey or Ebony? Why?

2. How are the acute and obtuse angles of the trapezoid related?

3. If Joey pushes one of the right angles through the two points, what type of figure is created? How does this relate to the major and minor arcs created above?

Lesson 4: Experiments with Inscribed Angles 1

^dKZzK&&hEd/KE^ Lesson 5 M5 GEOMETRY

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Lesson 5: Inscribed Angle Theorem and Its Applications

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has a measure of ͳͷ degrees, find the values of ݔ and ݕ. Explain how you ܤ The center of the circle below is ܱ. If angle know.

Lesson 5: Inscribed Angle Theorem and Its Applications 1

^dKZzK&&hEd/KE^ Lesson 6 M5 GEOMETRY

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Lesson 6: Unknown Angle Problems with Inscribed Angles in Circles

Exit Ticket

.Find the measure of angles ݔ and ݕ. Explain the relationships and theorems used

Lesson 6: Unknown Angle Problems with Inscribed Angles in Circles 1

^dKZzK&&hEd/KE^ Lesson 7 M5 GEOMETRY

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Lesson 7: The Angle Measure of an Arc

Exit Ticket

Given circle ܣ with diameters തܤܥതതത and തܦܧതതത and ݉ܥ෢ܦ ൌ ͷ͸ι. a. Name a central angle.

b. Name an inscribed angle.

c. Name a chord that is not a diameter.

?ܦܣܥס d. What is the measure of

?ܦܤܥס e. What is the measure of

f. Name ͵ angles of equal measure.

g. What is the degree measure of ܥܦ෣ܤ?

Lesson 7: The Angle Measure of an Arc 1

^dKZzK&&hEd/KE^ Lesson 8 M5 GEOMETRY

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Lesson 8: Arcs and Chords

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1. Given circle ܣ with radius ͳͲ, prove ܤܧ ൌ ܦܥ.

2. Given the circle at right, find ݉ܤ෢ܦ.

Lesson 8: Arcs and Chords 1

^dKZzK&&hEd/KE^ Lesson 9 M5 GEOMETRY

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Lesson 9: Arc Length and Areas of Sectors

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1. Find the arc length of ܴܲܳ෣.

2. Find the area of sector ܱܴܲ.

Lesson 9: Arc Length and Areas of Sectors 1

^dKZzK&&hEd/KE^ Lesson 10 M5 GEOMETRY

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Lesson 10: Unknown Length and Area Problems

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1. Given circle ܣ, find the following (round to the nearest hundredth). a. ݉ܤ෢ܥ in degrees

b. Area of sector ܤܣܥ

2. Find the shaded area (round to the nearest hundredth).

͸ʹι

Lesson 10: Unknown Length and Area Problems 1

^dKZzK&&hEd/KE^ Lesson 11 M5 GEOMETRY

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Lesson 11: Properties of Tangents

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1. If ܤܥ ൌ ͻ, ܣܤ ൌ ͸, and ܣܥ ൌ ͳͷ, is ܤܥശሬሬሬሬԦ tangent to circle ܣ? Explain.

2. Construct a line tangent to circle ܣ through point ܤ.

Lesson 11: Properties of Tangents 1

^dKZzK&&hEd/KE^ Lesson 12 M5 GEOMETRY

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Lesson 12: Tangent Segments

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1. Draw a circle tangent to both rays of this angle.

Explain how you .ܤܥܣס and ܥܤܣס be the points of tangency of your circle. Find the measures of ܥ and ܤ Let .2 determined your answer.

3. Let ܲ be the center of your circle. Find the measures of the angles in ᇞܣܲܤ.

Lesson 12: Tangent Segments 1

^dKZzK&&hEd/KE^ Lesson 13 M5 GEOMETRY

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Lesson 13: The Inscribed Angle Alternate—A Tangent Angle

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Find ܽ, ܾ, and ܿ.

Lesson 13: The Inscribed Angle Alternate—A Tangent Angle 1

^dKZzK&&hEd/KE^ Lesson 14 M5 GEOMETRY

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Lesson 14: Secant Lines; Secant Lines That Meet Inside a Circle

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ͳ ൌ ሺͳʹ͵ሻ ൌ ͸ͳι because it is half of the intercepted arc. Sandra says that you cannot ܥܨܦס݉ Lowell says that .1 ʹ ?because you do not have enough information. Who is correct and why ܥܨܦס determine the measure of

.ൌ ͻͻι, find and explain how you determined your answer ܥܨܧס݉ If .2 ܧܨܤס݉ .a

b. ݉ܤ෢ܧ

Lesson 14: Secant Lines; Secant Lines That Meet Inside a Circle 1

^dKZzK&&hEd/KE^ Lesson 15 M5 GEOMETRY

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Lesson 15: Secant Angle Theorem, Exterior Case

Exit Ticket

.Find ݔι . Explain your answer .1

.ൌݕιെݔι. Justify your work ܩ෢ܨ݉ ൌݕι൅ݔι and ܧ෢ܦ݉ Use the diagram to show that .2

Lesson 15: Secant Angle Theorem, Exterior Case 1

^dKZzK&&hEd/KE^ Lesson 16 M5 GEOMETRY

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Lesson 16: Similar Triangles in Circle-Secant (or Circle-Secant- Tangent) Diagrams

Exit Ticket

In the circle below, ݉ܩ෢ܨ ൌ ͵Ͳι, ݉ܦ෢ܧ ൌ ͳʹͲι, ܥܩ ൌ ͸, ܩܪ ൌ ʹ, ܨܪ ൌ ͵, ܥܨ ൌ Ͷ, ܪܧ ൌ ͻ, and ܨܧ ൌ ͳʹ.

.(ܧܪܦס݉) ܽ a. Find

.and explain your answer ,(ܧܥܦס݉) ܾ b. Find

.and explain your answer ,(ܦܪ) c. Find ݔ

.(ܩܦ) d. Find ݕ

Lesson 16: Similar Triangles in Circle-Secant (or Circle-Secant-Tangent) Diagrams 1

^dKZzK&&hEd/KE^ Lesson 17 M5 GEOMETRY

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Lesson 17: Writing the Equation for a Circle

Exit Ticket

.Describe the circle given by the equation ሺݔെ͹ሻଶ ൅ ሺݕെͺሻଶ ൌͻ .1

2. Write the equation for a circle with center ሺͲǡ െͶሻ and radius ͺ.

3. Write the equation for the circle shown below.

4. A circle has a diameter with endpoints at ሺ͸ǡ ͷሻ and ሺͺǡ ͷሻ. Write the equation for the circle.

Lesson 17: Writing the Equation for a Circle 1

^dKZzK&&hEd/KE^ Lesson 18 M5 GEOMETRY

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Lesson 18: Recognizing Equations of Circles

Exit Ticket

1. The graph of the equation below is a circle. Identify the center and radius of the circle. ݔଶ ൅ͳͲݔ൅ݕଶ െͺݕെͺൌͲ

2. Describe the graph of each equation. Explain how you know what the graph will look like. a. ݔଶ ൅ʹݔ൅ݕଶ ൌെͳ

͵b. ݔଶ ൅ݕଶ ൌെ

c. ݔଶ ൅ݕଶ ൅͸ݔ൅͸ݕൌ͹

Lesson 18: Recognizing Equations of Circles 1

^dKZzK&&hEd/KE^ Lesson 19 M5 GEOMETRY

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Lesson 19: Equations for Tangent Lines to Circles

Exit Ticket

.Consider the circle ሺݔ ൅ ʹሻଶ ൅ሺݕെ͵ሻଶ ൌͻ. There are two lines tangent to this circle having a slope of െͳ 1. Find the coordinates of the two points of tangency.

2. Find the equations of the two tangent lines.

Lesson 19: Equations for Tangent Lines to Circles 1

^dKZzK&&hEd/KE^ Lesson 20 M5 GEOMETRY

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Lesson 20: Cyclic Quadrilaterals

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.What value of ݔ guarantees that the quadrilateral shown in the diagram below is cyclic? Explain .1

,ൌ ʹͶ ܪܰ ൌ ͺ, and ܰܩ ,ൌ Ͷͺ ܬܰ ,ൌ Ͷ ܰܭ ,ൌ ͸Ͳι ܬܰܪס݉ ,ൌ ͳͺͲι ܬܪܭס݉ ൅ ܬܩܭס݉ ,ܬܪܭܩ Given quadrilateral .2 find the area of quadrilateral ܩܭܪܬ. Justify your answer.

Lesson 20: Cyclic Quadrilaterals 1

^dKZzK&&hEd/KE^ Lesson 21 M5 GEOMETRY

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Lesson 21: Ptolemy’s Theorem

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What is the length of the chord ܣܥതതതത? Explain your answer.

Lesson 21: Ptolemy’s Theorem 1