lesson 1: why move things around?
TRANSCRIPT
Lesson 1: Why Move Things Around? Date: 9/17/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 1
Lesson 1: Why Move Things Around?
Classwork
Exploratory Challenge 1
1. Describe, intuitively, what kind of transformation will be required to move the figure on the left to each of the figures (1β3) on the right. To help with this exercise, use a transparency to copy the figure on the left. Note that you are supposed to begin by moving the left figure to each of the locations in (1), (2), and (3).
Lesson 1: Why Move Things Around? Date: 9/17/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 1
2. Given two segments π΄π΅ and πΆπ·, which could be very far apart, how can we find out if they have the same length without measuring them individually? Do you think they have the same length? How do you check? In other words, why do you think we need to move things around on the plane?
Lesson 1: Why Move Things Around? Date: 9/17/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 1
Problem Set 1. Using as much of the new vocabulary as you can, try to describe what you see in the diagram below.
2. Describe, intuitively, what kind of transformation will be required to move Figure A on the left to its image on the right.
Lesson Summary
A transformation of the plane, to be denoted by πΉ, is a rule that assigns to each point π of the plane, one and only one (unique) point which will be denoted by πΉ(π).
So, by definition, the symbol πΉ(π) denotes a specific single point. The symbol πΉ(π) shows clearly that πΉ moves π to πΉ(π)
The point πΉ(π) will be called the image of π by πΉ
We also say πΉ maps π to πΉ(π)
If given any two points π and π, the distance between the images πΉ(π) and πΉ(π) is the same as the distance between the original points π and π, then the transformation πΉ preserves distance, or is distance-preserving.
A distance-preserving transformation is called a rigid motion (or an isometry), and the name suggests that it βmovesβ the points of the plane around in a βrigidβ fashion.
A
B
F(A)
F(B)
Figure A
Image of A
Lesson 2: Definition of Translation and Three Basic Properties Date: 9/17/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 2
Lesson 2: Definition of Translation and Three Basic Properties
Classwork
Exercise 1
Draw at least three different vectors and show what a translation of the plane along each vector will look like. Describe what happens to the following figures under each translation, using appropriate vocabulary and notation as needed.
Lesson 2: Definition of Translation and Three Basic Properties Date: 9/17/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 2
Exercise 2
The diagram below shows figures and their images under a translation along π»πΌοΏ½οΏ½οΏ½οΏ½β . Use the original figures and the translated images to fill in missing labels for points and measures.
Lesson 2: Definition of Translation and Three Basic Properties Date: 9/17/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 2
Problem Set 1. Translate the plane containing Figure A along π΄π΅οΏ½οΏ½οΏ½οΏ½οΏ½β . Use your
transparency to sketch the image of Figure A by this translation. Mark points on Figure A and label the image of Figure A accordingly.
2. Translate the plane containing Figure B along π΅π΄οΏ½οΏ½οΏ½οΏ½οΏ½β Use your transparency to sketch the image of Figure B by this translation. Mark points on Figure B and label the image of Figure B accordingly.
Lesson Summary
Translation occurs along a given vector:
A vector is a segment in the plane. One of its two endpoints is known as a starting point; while the other is known simply as the endpoint.
The length of a vector is, by definition, the length of its underlying segment.
Pictorially note the starting and endpoints:
A translation of a plane along a given vector is a basic rigid motion of a plane.
The three basic properties of translation are:
(T1) A translation maps a line to a line, a ray to a ray, a segment to a segment, and an angle to an angle.
(T2) A translation preserves lengths of segments.
(T3) A translation preserves degrees of angles.
Lesson 2: Definition of Translation and Three Basic Properties Date: 9/17/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 2
3. Draw an acute angle (your choice of degree), a segment with length 3 cm, a point, a circle with radius 1 in, and a vector (your choice of length, i.e., starting point and ending point). Label points and measures (measurements do not need to be precise, but your figure must be labeled correctly). Use your transparency to translate all of the figures youβve drawn along the vector. Sketch the images of the translated figures and label them.
4. What is the length of the translated segment? How does this length compare to the length of the original segment?
Explain.
5. What is the length of the radius in the translated circle? How does this radius length compare to the radius of the original circle? Explain.
6. What is the degree of the translated angle? How does this degree compare to the degree of the original angle? Explain.
7. Translate point π· along vector π΄π΅οΏ½οΏ½οΏ½οΏ½οΏ½β and label the image π·β². What do you notice about the line containing vector π΄π΅οΏ½οΏ½οΏ½οΏ½οΏ½β , and the line containing points π· and π·β²? (Hint: Will the lines ever intersect?)
8. Translate point πΈ along vector π΄π΅οΏ½οΏ½οΏ½οΏ½οΏ½β and label the image πΈβ². What do you notice about the line containing vector π΄π΅οΏ½οΏ½οΏ½οΏ½οΏ½β , and the line containing points πΈ and πΈβ²?
Lesson 3: Translating Lines Date: 9/17/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 3
Lesson 3: Translating Lines
Classwork
Exercises
1. Draw a line passing through point P that is parallel to line πΏ. Draw a second line passing through point π that is parallel to line πΏ, that is distinct (i.e., different) from the first one. What do you notice?
2. Translate line πΏ along the vector π΄π΅οΏ½οΏ½οΏ½οΏ½οΏ½β . What do you notice about πΏ and its image πΏβ²?
Lesson 3: Translating Lines Date: 9/17/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 3
3. Line πΏ is parallel to vector π΄π΅οΏ½οΏ½οΏ½οΏ½οΏ½β . Translate line πΏ along vector π΄π΅οΏ½οΏ½οΏ½οΏ½οΏ½β . What do you notice about πΏ and its image, πΏβ²?
4. Translate line πΏ along the vector π΄π΅οΏ½οΏ½οΏ½οΏ½οΏ½β . What do you notice about πΏ and its image, πΏβ²?
Lesson 3: Translating Lines Date: 9/17/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 3
5. Line πΏ has been translated along vector π΄π΅οΏ½οΏ½οΏ½οΏ½οΏ½β resulting in πΏβ. What do you know about lines πΏ and πΏβ?
6. Translate πΏ1 and πΏ2 along vector π·πΈοΏ½οΏ½οΏ½οΏ½οΏ½β . Label the images of the lines. If lines πΏ1 and πΏ2 are parallel, what do you know about their translated images?
Lesson 3: Translating Lines Date: 9/17/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 3
Problem Set 1. Translate β πππ, point π΄, point π΅, and rectangle π»πΌπ½πΎ along vector πΈπΉοΏ½οΏ½οΏ½οΏ½οΏ½β Sketch the images and label all points using
prime notation.
2. What is the measure of the translated image of β πππ. How do you know?
3. Connect π΅ to π΅β². What do you know about the line formed by π΅π΅β² and the line containing the vector πΈπΉοΏ½οΏ½οΏ½οΏ½οΏ½β ?
4. Connect π΄ to π΄β². What do you know about the line formed by π΄π΄β² and the line containing the vector πΈπΉοΏ½οΏ½οΏ½οΏ½οΏ½β ?
5. Given that figure π»πΌπ½πΎ is a rectangle, what do you know about lines π»πΌ and π½πΎ and their translated images? Explain.
Lesson Summary
Two lines are said to be parallel if they do not intersect.
Translations map parallel lines to parallel lines.
Given a line L and a point π not lying on πΏ, there is at most one line passing through π and parallel to πΏ.
Lesson 4: Definition of Reflection and Basic Properties Date: 9/17/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 4
Lesson 4: Definition of Reflection and Basic Properties
Classwork
Exercises
1. Refect βπ΄π΅πΆ and Figure π· across line πΏ. Label the reflected images.
2. Which figure(s) were not moved to a new location on the plane under this transformation?
Lesson 4: Definition of Reflection and Basic Properties Date: 9/17/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 4
3. Reflect the images across line πΏ. Label the reflected images.
4. Answer the questions about the image above.
a. Use a protractor to measure the reflected β π΄π΅πΆ.
b. Use a ruler to measure the length of image of πΌπ½ after the reflection.
Lesson 4: Definition of Reflection and Basic Properties Date: 9/17/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 4
5. Reflect Figure R and βπΈπΉπΊ across line πΏ. Label the reflected images.
Basic Properties of Reflections:
(Reflection 1) A reflection maps a line to a line, a ray to a ray, a segment to a segment, and an angle to an angle.
(Reflection2) A reflection preserves lengths of segments.
(Reflection 3) A reflection preserves degrees of angles.
If the reflection is across a line L and P is a point not on L, then L bisects the segment PPβ, joining P to its reflected image Pβ. That is, the lengths of OP and OPβ are equal.
Lesson 4: Definition of Reflection and Basic Properties Date: 9/17/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 4
Use the picture below for Exercises 6β9.
6. Use the picture to label the unnamed points.
7. What is the measure of β π½πΎπΌ? β πΎπΌπ½? β π΄π΅πΆ? How do you know?
8. What is the length of segment π ππππππ‘πππ(πΉπ»)? πΌπ½? How do you know?
9. What is the location of π ππππππ‘πππ(π·)? Explain.
Lesson 4: Definition of Reflection and Basic Properties Date: 9/17/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 4
Problem Set 1. In the picture below, β π·πΈπΉ = 56Β°, β π΄πΆπ΅ = 114Β°, π΄π΅ = 12.6 π’πππ‘π , π½πΎ = 5.32 π’πππ‘π , point πΈ is on line πΏ and
point πΌ is off of line πΏ. Let there be a reflection across line πΏ. Reflect and label each of the figures, and answer the questions that follow.
Lesson Summary
A reflection is another type of basic rigid motion.
Reflections occur across lines. The line that you reflect across is called the line of reflection.
When a point, π, is joined to its reflection, πβ², the line of reflection bisects the segment, ππβ².
Lesson 4: Definition of Reflection and Basic Properties Date: 9/17/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 4
2. What is the size of π ππππππ‘πππ(β π·πΈπΉ)? Explain.
3. What is the length of π ππππππ‘πππ(π½πΎ)? Explain.
4. What is the size of π ππππππ‘πππ(β π΄πΆπ΅)?
5. What is the length of π ππππππ‘πππ(π΄π΅)?
6. Two figures in the picture were not moved under the reflection. Name the two figures and explain why they were not moved.
7. Connect points πΌ and πΌβ Name the point of intersection of the segment with the line of reflection point π. What do you know about the lengths of segments πΌπ and ππΌβ?
Lesson 5: Definition of Rotation and Basic Properties Date: 9/17/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 5
Lesson 5: Definition of Rotation and Basic Properties
Classwork
Example 1
Let there be a rotation around center π,π degrees.
If π = 30, then the plane moves as shown:
If π = β30, then the plane moves as shown:
Exercises
1. Let there be a rotation of π degrees around center π. Let π be a point other than π. Select a π so that π β₯ 0. Find πβ (i.e., the rotation of point π) using a transparency.
Lesson 5: Definition of Rotation and Basic Properties Date: 9/17/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 5
2. Let there be a rotation of d degrees around center π. Let π be a point other than π. Select a π so that π < 0. Find πβ (i.e., the rotation of point π) using a transparency.
3. Which direction did the point π rotate when π β₯ 0?
4. Which direction did the point π rotate when π < 0?
Lesson 5: Definition of Rotation and Basic Properties Date: 9/17/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 5
5. Let L be a line, π΄π΅οΏ½οΏ½οΏ½οΏ½οΏ½β be a ray, πΆπ· be a segment, and β πΈπΉπΊ be an angle, as shown. Let there be a rotation of π degrees around point π. Find the images of all figures when π β₯ 0.
Lesson 5: Definition of Rotation and Basic Properties Date: 9/17/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 5
6. Let π΄π΅ be a segment of length 4 units and β πΆπ·πΈ be an angle of size 45Λ. Let there be a rotation by π degrees, where π < 0, about π. Find the images of the given figures. Answer the questions that follow.
a. What is the length of the rotated segment π ππ‘ππ‘πππ(π΄π΅)?
b. What is the degree of the rotated angle π ππ‘ππ‘πππ (β πΆπ·πΈ)?
Lesson 5: Definition of Rotation and Basic Properties Date: 9/17/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 5
7. Let πΏ1, πΏ2 be parallel lines. Let there be a rotation by d degrees, where β360 < π < 360, about π. Is (πΏ1)β² β₯ (πΏ2)β²?
8. Let πΏ be a line and π be the center of rotation. Let there be a rotation by π degrees, where π β 180 about π. Are the lines πΏ and πΏβ parallel?
Lesson 5: Definition of Rotation and Basic Properties Date: 9/17/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 5
Problem Set 1. Let there be a rotation by β 90Λ around the center π.
Lesson Summary
Rotations require information about the center of rotation and the degree in which to rotate. Positive degrees of rotation move the figure in a counterclockwise direction. Negative degrees of rotation move the figure in a clockwise direction.
Basic Properties of Rotations:
(R1) A rotation maps a line to a line, a ray to a ray, a segment to a segment, and an angle to an angle.
(R2) A rotation preserves lengths of segments.
(R3) A rotation preserves degrees of angles.
When parallel lines are rotated, their images are also parallel. A line is only parallel to itself when rotated exactly 180Λ.
Lesson 5: Definition of Rotation and Basic Properties Date: 9/17/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 5
2. Explain why a rotation of 90 degrees never maps a line to a line parallel to itself.
3. A segment of length 94 cm has been rotated π degrees around a center π. What is the length of the rotated segment? How do you know?
4. An angle of size 124Λ has been rotated π degrees around a center π. What is the size of the rotated angle? How do
you know?
Lesson 6: Rotations of 180 Degrees Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 6
Lesson 6: Rotations of 180 Degrees
Classwork
Example 1
The picture below shows what happens when there is a rotation of 180Λ around center π.
Example 2
The picture below shows what happens when there is a rotation of 180Λ around center π, the origin of the coordinate plane.
Lesson 6: Rotations of 180 Degrees Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 6
Exercises
1. Using your transparency, rotate the plane 180 degrees, about the origin. Let this rotation be π ππ‘ππ‘πππ0. What are the coordinates of π ππ‘ππ‘πππ0(2,β4)?
2. Let π ππ‘ππ‘πππ0 be the rotation of the plane by 180 degrees, about the origin. Without using your transparency, find π ππ‘ππ‘πππ0(β3, 5).
Lesson 6: Rotations of 180 Degrees Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 6
3. Let π ππ‘ππ‘πππ0 be the rotation of 180 degrees around the origin. Let πΏ be the line passing through (β6, 6) parallel to the π₯-axis. Find π ππ‘ππ‘πππ0(πΏ). Use your transparency if needed.
4. Let π ππ‘ππ‘πππ0 be the rotation of 180 degrees around the origin. Let πΏ be the line passing through (7,0) parallel to the π¦-axis. Find π ππ‘ππ‘πππ0(πΏ). Use your transparency if needed.
Lesson 6: Rotations of 180 Degrees Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 6
5. Let π ππ‘ππ‘πππ0 be the rotation of 180 degrees around the origin. Let πΏ be the line passing through (0,2) parallel to the π₯-axis. Is πΏ parallel to π ππ‘ππ‘πππ0(πΏ)?
6. Let π ππ‘ππ‘πππ0 be the rotation of 180 degrees around the origin. Let πΏ be the line passing through (4,0) parallel to the π¦-axis. Is πΏ parallel to π ππ‘ππ‘πππ0(πΏ)?
Lesson 6: Rotations of 180 Degrees Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 6
7. Let π ππ‘ππ‘πππ0 be the rotation of 180 degrees around the origin. Let πΏ be the line passing through (0,β1) parallel to the π₯-axis. Is πΏ parallel to π ππ‘ππ‘πππ0(πΏ)?
8. Let π ππ‘ππ‘πππ0 be the rotation of 180 degrees around the origin. Is πΏ parallel to π ππ‘ππ‘πππ0(πΏ)? Use your transparency if needed.
Lesson 6: Rotations of 180 Degrees Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 6
9. Let π ππ‘ππ‘πππ0 be the rotation of 180 degrees around the origin. Is πΏ parallel to π ππ‘ππ‘πππ0(πΏ)? Use your transparency if needed.
Lesson 6: Rotations of 180 Degrees Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 6
Problem Set Use the following diagram for problems 1β5. Use your transparency, as needed.
1. Looking only at segment π΅πΆ, is it possible that a 180Λ rotation would map π΅πΆ onto π΅β²πΆβ²? Why or why not?
2. Looking only at segment π΄π΅, is it possible that a 180Λ rotation would map π΄π΅ onto π΄β²π΅β²? Why or why not? 3. Looking only at segment π΄πΆ, is it possible that a 180Λ rotation would map π΄πΆ onto π΄β²πΆβ²? Why or why not?
4. Connect point π΅ to point π΅β², point πΆ to point πΆβ², and point π΄ to point π΄β². What do you notice? What do you think that point is?
5. Would a rotation map triangle π΄π΅πΆ onto triangle π΄β²π΅β²πΆβ²? If so, define the rotation (i.e., degree and center). If not, explain why not.
Lesson Summary
A rotation of 180 degrees around π is the rigid motion so that if π is any point in the plane, π,π and π ππ‘ππ‘πππ(π) are collinear (i.e., lie on the same line).
Given a 180-degree rotation, π 0 around the origin π of a coordinate system, and a point π with coordinates (π, π), it is generally said that π 0(π) is the point with coordinates (βπ,βπ).
Theorem. Let π be a point not lying on a given line πΏ. Then the 180-degree rotation around π maps πΏ to a line parallel to πΏ.
Lesson 6: Rotations of 180 Degrees Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 6
6. The picture below shows right triangles π΄π΅πΆ and π΄β²π΅β²πΆβ², where the right angles are at π΅ and π΅β². Given that π΄π΅ = π΄β²π΅β² = 1, and π΅πΆ = π΅β²πΆβ² = 2, π΄π΅ is not parallel to π΄β²π΅β², is there a 180Λ rotation that would map βπ΄π΅πΆ onto βπ΄β²π΅β²πΆβ²? Explain.
Lesson 7: Sequencing Translations Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 7
Lesson 7: Sequencing Translations
Classwork
Exploratory Challenge
1.
a. Translate β π΄π΅πΆ and segment πΈπ· along vector πΉπΊοΏ½οΏ½οΏ½οΏ½οΏ½β . Label the translated images appropriately, i.e., π΄β²π΅β²πΆβ² and πΈβ²π·β².
b. Translate β π΄β²π΅β²πΆβ²and segment πΈβ²π·β² along vector π»πΌοΏ½οΏ½οΏ½οΏ½β . Label the translated images appropriately, i.e., π΄β²β²π΅β²β²πΆβ²β² and πΈβ²β²π·β²β².
c. How does the size of β π΄π΅πΆ compare to the size of β π΄β²β²π΅β²β²πΆβ²β²?
d. How does the length of segment πΈπ· compare to the length of the segment πΈβ²β²π·β²β²?
e. Why do you think what you observed in parts (d) and (e) were true?
Lesson 7: Sequencing Translations Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 7
2. Translate βπ΄π΅πΆ along vector πΉπΊοΏ½οΏ½οΏ½οΏ½οΏ½β and then translate its image along vector π½πΎοΏ½οΏ½οΏ½οΏ½β . Be sure to label the images appropriately.
Lesson 7: Sequencing Translations Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 7
3. Translate figure π΄π΅πΆπ·πΈπΉ along vector πΊπ»οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½β . Then translate its image along vector π½πΌοΏ½οΏ½οΏ½β . Label each image appropriately.
Lesson 7: Sequencing Translations Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 7
4.
a. Translate Circle π΄ and Ellipse πΈ along vector π΄π΅οΏ½οΏ½οΏ½οΏ½οΏ½β . Label the images appropriately.
b. Translate Circle π΄β² and Ellipse πΈβ² along vector πΆπ·οΏ½οΏ½οΏ½οΏ½οΏ½β . Label each image appropriately.
c. Did the size or shape of either figure change after performing the sequence of translations? Explain.
Lesson 7: Sequencing Translations Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 7
5. The picture below shows the translation of Circle A along vector πΆπ·οΏ½οΏ½οΏ½οΏ½οΏ½β . Name the vector that will map the image of Circle A back onto itself.
6. If a figure is translated along vector ππ οΏ½οΏ½οΏ½οΏ½οΏ½β , what translation takes the figure back to its original location?
Lesson 7: Sequencing Translations Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 7
Problem Set 1. Sequence translations of Parallelogram π΄π΅πΆπ· (a quadrilateral in which both pairs of opposite sides are parallel)
along vectors π»πΊοΏ½οΏ½οΏ½οΏ½οΏ½οΏ½β and πΉπΈοΏ½οΏ½οΏ½οΏ½οΏ½β . Label the translated images.
2. What do you know about π΄π· and π΅πΆ compared with π΄β²π·β² and π΅β²πΆβ²? Explain.
3. Are π΄β²π΅β² and π΄β²β²π΅β²β² equal in length? How do you know?
4. Translate the curved shape π΄π΅πΆ along the given vector. Label the image.
Lesson Summary
Translating a figure along one vector then translating its image along another vector is an example of a sequence of transformations.
A sequence of translations enjoys the same properties as a single translation. Specifically, the figuresβ lengths and degrees of angles are preserved.
If a figure undergoes two transformations, πΉ and πΊ, and is in the same place it was originally, then the figure has been mapped onto itself.
Lesson 7: Sequencing Translations Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 7
5. What vector would map the shape π΄β²π΅β²πΆβ² back onto π΄π΅πΆ?
Lesson 8: Sequencing Reflections and Translations Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 8
Lesson 8: Sequencing Reflections and Translations
Classwork
Exercises
Use the figure below to answer Exercises 1β3.
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 8
1. Figure π΄ was translated along vector π΅π΄οΏ½οΏ½οΏ½οΏ½οΏ½β resulting in πππππ πππ‘πππ(πΉπππ’ππ π΄). Describe a sequence of translations that would map Figure π΄ back onto its original position.
2. Figure π΄ was reflected across line πΏ resulting in π ππππππ‘πππ(πΉπππ’ππ π΄). Describe a sequence of reflections that would map Figure π΄ back onto its original position.
3. Can πππππ πππ‘ππππ΅π΄οΏ½οΏ½οΏ½οΏ½οΏ½β undo the transformation of πππππ πππ‘ππππ·πΆοΏ½οΏ½οΏ½οΏ½οΏ½β ? Why or why not?
Lesson 8: Sequencing Reflections and Translations Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 8
Use the diagram below for Exercises 4- 5.
4. Let there be the translation along vectorπ΄π΅οΏ½οΏ½οΏ½οΏ½οΏ½β and a reflection across line πΏ. Let the black figure above be figure π. Use a transparency to perform the following sequence: Translate figure π, then reflect figure S. Label the image πβ².
5. Let there be the translation along vectorπ΄π΅οΏ½οΏ½οΏ½οΏ½οΏ½β and a reflection across line πΏ. Let the black figure above be figure π. Use a transparency to perform the following sequence: Reflect figure π, then translate figure S. Label the image πβ²β²
Lesson 8: Sequencing Reflections and Translations Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 8
6. Using your transparency, show that under a sequence of any two translations, πππππ πππ‘πππ and πππππ πππ‘πππ0 (along different vectors), that the sequence of the πππππ πππ‘πππ followed by the πππππ πππ‘πππ0 is equal to the sequence of the πππππ πππ‘πππ0 followed by the πππππ πππ‘πππ. That is, draw a figure, π΄, and two vectors. Show that the translation along the first vector, followed by a translation along the second vector, places the figure in the same location as when you perform the translations in the reverse order. (This fact will be proven in high school). Label the transformed image π΄β². Now draw two new vectors and translate along them just as before. This time label the transformed image π΄β²β². Compare your work with a partner. Was the statement that βthe sequence of the πππππ πππ‘πππ followed by the πππππ πππ‘πππ0 is equal to the sequence of the πππππ πππ‘πππ0 followed by the πππππ πππ‘πππβ true in all cases? Do you think it will always be true?
7. Does the same relationship you noticed in Exercise 6 hold true when you replace one of the translations with a reflection. That is, is the following statement true: A translation followed by a reflection is equal to a reflection followed by a translation?
Lesson 8: Sequencing Reflections and Translations Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 8
Problem Set 1. Let there be a reflection across line πΏ and let there be a translation along vector π΄π΅οΏ½οΏ½οΏ½οΏ½οΏ½β as shown. If π denotes the
black figure, compare the translation of π followed by the reflection of π with the reflection of π followed by the translation of π.
2. Let πΏ1 and πΏ2 be parallel lines and let π ππππππ‘πππ1 and π ππππππ‘πππ2 be the reflections across πΏ1 and πΏ2, respectively (in that order). Show that a π ππππππ‘πππ2 followed by π ππππππ‘πππ1 is not equal to a π ππππππ‘πππ1 followed by π ππππππ‘πππ2. (Hint: Take a point on πΏ1 and see what each of the sequences does to it.)
3. Let πΏ1 and πΏ2 be parallel lines and let π ππππππ‘πππ1 and π ππππππ‘πππ2 be the reflections across πΏ1 and πΏ2, respectively (in that order). Can you guess what π ππππππ‘πππ1 followed by π ππππππ‘πππ2 is? Give as persuasive an argument as you can. (Hint: Examine the work you just finished for the last problem.)
Lesson Summary
A reflection across a line followed by a reflection across the same line places all figures in the plane back onto their original position.
A reflection followed by a translation does not place a figure in the same location in the plane as a translation followed by a reflection. The order in which we perform a sequence of rigid motions matters.
Lesson 9: Sequencing Rotations Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 9
Lesson 9: Sequencing Rotations
Classwork
Exploratory Challenge
1.
a. Rotate β³ π΄π΅πΆ d degrees around center π·. Label the rotated image as β³ π΄β²π΅β²πΆβ².
b. Rotate β³ π΄β²π΅β²πΆβ² d degrees around center πΈ. Label the rotated image as β³ π΄β²β²π΅β²β²πΆβ²β².
c. Measure and label the angles and side lengths of β³ π΄π΅πΆ. How do they compare with the images β³ π΄β²π΅β²πΆβ² and β³ π΄β²β²π΅β²β²πΆβ²β²?
d. How can you explain what you observed in part (c)? What statement can you make about properties of sequences of rotations as they relate to a single rotation?
Lesson 9: Sequencing Rotations Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 9
2.
a. Rotate β³ π΄π΅πΆ d degrees around center π· and then rotate again π degrees around center πΈ. Label the image as β³ π΄β²π΅β²πΆβ² after you have completed both rotations.
b. Can a single rotation around center π· map β³ π΄β²π΅β²πΆβ² onto β³ π΄π΅πΆ?
c. Can a single rotation around center πΈ map β³ π΄β²π΅β²πΆβ² onto β³ π΄π΅πΆ?
d. Can you find a center that would map β³ π΄β²π΅β²πΆβ² onto β³ π΄π΅πΆ in one rotation? If so, label the center πΉ.
Lesson 9: Sequencing Rotations Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 9
3.
a. Rotate β³ π΄π΅πΆ 90Λ (counterclockwise) around center π· then rotate the image another 90Λ (counterclockwise around center πΈ. Label the image β³ π΄β²π΅β²πΆβ².
b. Rotate β³ π΄π΅πΆ 90Λ (counterclockwise) around center πΈ then rotate the image another 90Λ (counterclockwise around center π·. Label the image β³ π΄β²β²π΅β²β²πΆβ²β².
c. What do you notice about the locations of β³ π΄β²π΅β²πΆβ² and β³ π΄β²β²π΅β²β²πΆβ²β²? Does the order in which you rotate a figure around different centers have an impact on the final location of the figures image?
Lesson 9: Sequencing Rotations Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 9
4.
a. Rotate β³ π΄π΅πΆ 90Λ (counterclockwise) around center D then rotate the image another 45Λ (counterclockwise) around center D. Label theβ³ π΄β²π΅β²πΆβ².
b. Rotate β³ π΄π΅πΆ 45Λ (counterclockwise) around center D then rotate the image another 90Λ (counterclockwise) around center D. Label theβ³ π΄β²β²π΅β²β²πΆβ²β².
c. What do you notice about the locations of β³ π΄β²π΅β²πΆβ² and β³ π΄β²β²π΅β²β²πΆβ²β²? Does the order in which you rotate a figure around the same center have an impact on the final location of the figures image?
Lesson 9: Sequencing Rotations Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 9
5. In the figure below, β³ π΄π΅πΆ has been rotated around two different centers and its image is β³ π΄β²π΅β²πΆβ². Describe a sequence of rigid motions that would map β³ π΄π΅πΆ onto β³ π΄β²π΅β²πΆβ².
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 9
Problem Set 1. Refer to the figure below.
a. Rotate β π΄π΅πΆ and segment π·πΈ π degrees around center πΉ, then π degrees around center πΊ. Label the final
location of the images as β π΄β²π΅β²πΆβ² and DβEβ.
b. What is the size of β π΄π΅πΆ and how does it compare to the size of β π΄β²π΅β²πΆβ²? Explain.
c. What is the length of segment DE and how does it compare to the length of segment π·β²πΈβ²? Explain.
Lesson Summary
Sequences of rotations have the same properties as a single rotation:
β’ A sequence of rotations preserves degrees of measures of angles.
β’ A sequence of rotations preserves lengths of segments. The order in which a sequence of rotations around different centers is performed matters with respect to
the final location of the image of the figure that is rotated.
The order in which a sequence of rotations around the same center is performed does not matter. The image of the figure will be in the same location.
Lesson 9: Sequencing Rotations Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 9
2. Refer to the figure given below.
a. Let π ππ‘ππ‘πππ1 be a counterclockwise rotation of 90Λ around the center π. Let π ππ‘ππ‘πππ2 be a clockwise rotation of (β45)Λ around the center π. Determine the approximate location of π ππ‘ππ‘πππ1(β³ π΄π΅πΆ) followed by π ππ‘ππ‘πππ2. Label the image of triangle π΄π΅πΆ as π΄β²π΅β²πΆβ².
b. Describe the sequence of rigid motions that would map β³ π΄π΅πΆ onto β³ π΄β²π΅β²πΆβ².
3. Refer to the figure given below. Let π be a rotation of (β90)Λ around the center π. Let π ππ‘ππ‘πππ2 be a rotation of (β45)Λ around the same center π. Determine the approximate location of π ππ‘ππ‘πππ1(βπ΄π΅πΆ) followed by π ππ‘ππ‘πππ2(βπ΄π΅πΆ). Label the image of triangle π΄π΅πΆ as π΄β²π΅β²πΆβ².
Lesson 10: Sequences of Rigid Motions Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 10
Lesson 10: Sequences of Rigid Motions
Classwork
Exercises
1. In the following picture, triangle π΄π΅πΆ can be traced onto a transparency and mapped onto triangle π΄β²π΅β²πΆβ². Which basic rigid motion, or sequence of, would map one triangle onto the other?
2. In the following picture, triangle π΄π΅πΆ can be traced onto a transparency and mapped onto triangle π΄β²π΅β²πΆβ². Which basic rigid motion, or sequence of, would map one triangle onto the other?
.
Lesson 10: Sequences of Rigid Motions Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 10
3. In the following picture, triangle π΄π΅πΆ can be traced onto a transparency and mapped onto triangle π΄β²π΅β²πΆβ². Which basic rigid motion, or sequence of, would map one triangle onto the other?
4. In the following picture, we have two pairs of triangles. In each pair, triangle π΄π΅πΆ can be traced onto a transparency and mapped onto triangle π΄β²π΅β²πΆβ². Which basic rigid motion, or sequence of, would map one triangle onto the other? Scenario 1:
Scenario 2:
Lesson 10: Sequences of Rigid Motions Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 10
5. Let two figures π΄π΅πΆ and π΄β²π΅β²πΆβ² be given so that the length of curved segment π΄πΆ = the length of curved segment π΄β²πΆβ², |β π΅| = |β π΅β²| = 80Β°, and |π΄π΅| = |π΄β²π΅β²| = 5. With clarity and precision, describe a sequence of rigid motions that would map figure π΄π΅πΆ onto figure π΄β²π΅β²πΆβ².
Lesson 10: Sequences of Rigid Motions Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 10
Problem Set 1. Let there be the translation along vector οΏ½βοΏ½, let there be the rotation around point π΄, β90 degrees (clockwise), and
let there be the reflection across line πΏ. Let π be the figure as shown below. Show the location of π after performing the following sequence: a translation followed by a rotation followed by a reflection.
2. Would the location of the image of π in the previous problem be the same if the translation was performed first
instead of last, i.e., does the sequence: translation followed by a rotation followed by a reflection equal a rotation followed by a reflection followed by a translation? Explain.
Lesson 10: Sequences of Rigid Motions Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 10
3. Use the same coordinate grid, below, to complete parts (a)β(c).
a. Reflect triangle π΄π΅πΆ across the vertical line, parallel to the π¦-axis, going through point (1, 0). Label the
transformed points π΄π΅πΆ as π΄β²,π΅β²,πΆβ², respectively.
b. Reflect triangle π΄β²π΅β²πΆβ² across the horizontal line, parallel to the π₯-axis going through point (0,β1). Label the transformed points of π΄βπ΅βπΆβ as π΄β²β²π΅β²β²πΆβ²β², respectively.
c. Is there a single rigid motion that would map triangle π΄π΅πΆ to triangle π΄β²β²π΅β²β²πΆβ²β² ?
Lesson 11: Definition of Congruence and Some Basic Properties Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 11
Lesson 11: Definition of Congruence and Some Basic Properties
Classwork
Exercise 1
a. Describe the sequence of basic rigid motions that shows π1 β π2.
Lesson 11: Definition of Congruence and Some Basic Properties Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 11
b. Describe the sequence of basic rigid motions that shows π2 β π3.
Lesson 11: Definition of Congruence and Some Basic Properties Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 11
c. Describe the sequence of basic rigid motions that shows π1 β π3.
Lesson 11: Definition of Congruence and Some Basic Properties Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 11
Exercise 2
Perform the sequence of a translation followed by a rotation of Figure πππ, where π is a translation along a vector π΄π΅οΏ½οΏ½οΏ½οΏ½οΏ½β and π is a rotation of π degrees (you choose π) around a center π. Label the transformed figure πβ²πβ²πβ². Will πππ β πβ²πβ²πβ²?
Lesson 11: Definition of Congruence and Some Basic Properties Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 11
Problem Set 1. Given two right triangles with lengths shown below, is there one basic rigid motion that maps one to the other?
Explain.
2. Are the two right triangles shown below congruent? If so, describe the congruence that would map one triangle onto the other.
Lesson Summary
Given that sequences enjoy the same basic properties of basic rigid motions, we can state three basic properties of congruences:
(C1) A congruence maps a line to a line, a ray to a ray, a segment to a segment, and an angle to an angle.
(C2) A congruence preserves lengths of segments.
(C3) A congruence preserves degrees of angles.
The notation used for congruence is β .
Lesson 11: Definition of Congruence and Some Basic Properties Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 11
3.
a. Given two rays, ππ΄οΏ½οΏ½οΏ½οΏ½οΏ½β and πβ²π΄β²οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½β , describe the congruence that maps ππ΄οΏ½οΏ½οΏ½οΏ½οΏ½β to πβ²π΄β²οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½β .
b. Describe the congruence that maps πβ²π΄β²οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½β to ππ΄οΏ½οΏ½οΏ½οΏ½οΏ½β .
Lesson 12: Angles Associated with Parallel Lines Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 12
Lesson 12: Angles Associated with Parallel Lines
Classwork
Exploratory Challenge 1
In the figure below, πΏ1 is not parallel to πΏ2, and π is a transversal. Use a protractor to measure angles 1β8. Which, if any, are equal? Explain why. (Use your transparency, if needed).
Lesson 12: Angles Associated with Parallel Lines Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 12
Exploratory Challenge 2
In the figure below, πΏ1 β₯ πΏ2, and π is a transversal. Use a protractor to measure angles 1β8. List the angles that are equal in measure.
a. What did you notice about the measures of β 1 and β 5? Why do you think this is so? (Use your transparency, if needed).
b. What did you notice about the measures of β 3 and β 7? Why do you think this is so? (Use your transparency, if needed.) Are there any other pairs of angles with this same relationship? If so, list them.
c. What did you notice about the measures of β 4 and β 6? Why do you think this is so? (Use your transparency, if needed). Is there another pair of angles with this same relationship?
Lesson 12: Angles Associated with Parallel Lines Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 12
Problem Set Use the diagram below to do Problems 1β6.
Lesson Summary
Angles that are on the same side of the transversal in corresponding positions (above each of πΏ1 and πΏ2 or below each of πΏ1 and πΏ2) are called corresponding angles. For example, β 2 and β 4.
When angles are on opposite sides of the transversal and between (inside) the lines πΏ1 and πΏ2, they are called alternate interior angles. For example, β 3 and β 7.
When angles are on opposite sides of the transversal and outside of the parallel lines (above πΏ1 and below πΏ2), they are called alternate exterior angles. For example, β 1 and β 5.
When parallel lines are cut by a transversal, the corresponding angles, alternate interior angles, and alternate exterior angles are equal. If the lines are not parallel, then the angles are not equal.
Lesson 12: Angles Associated with Parallel Lines Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 12
1. Identify all pairs of corresponding angles. Are the pairs of corresponding angles equal in measure? How do you know?
2. Identify all pairs of alternate interior angles. Are the pairs of alternate interior angles equal in measure? How do you know?
3. Use an informal argument to describe why β 1 and β 8 are equal in measure if πΏ1 β₯ πΏ2.
4. Assuming πΏ1 β₯ πΏ2 if the measure of β 4 is 73Λ, what is the measure of β 8? How do you know?
5. Assuming πΏ1 β₯ πΏ2, if the measure of β 3 is 107Λ degrees, what is the measure of β 6? How do you know?
6. Assuming πΏ1 β₯ πΏ2, if the measure of β 2 is 107Λ, what is the measure of β 7? How do you know?
7. Would your answers to Problems 4β6 be the same if you had not been informed that πΏ1 β₯ πΏ2? Why or why not?
8. Use an informal argument to describe why β 1 and β 5 are equal in measure if πΏ1 β₯ πΏ2.
9. Use an informal argument to describe why β 4 and β 5 are equal in measure if πΏ1 β₯ πΏ2.
10. Assume that πΏ1 is not parallel to πΏ2. Explain why β 3 β β 7.
Lesson 13: Angle Sum of a Triangle Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 13
Lesson 13: Angle Sum of a Triangle
Classwork Concept Development
β 1 + β 2 + β 3 = β 4 + β 5 + β 6 = β 7 + β 8 + β 9 = 180
Note that the sum of angles 7 and 9 must equal 90Λ because of the known right angle in the right triangle.
Lesson 13: Angle Sum of a Triangle Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 13
Exploratory Challenge 1
Let triangle π΄π΅πΆ be given. On the ray from π΅ to πΆ, take a point π· so that πΆ is between π΅ and π·. Through point πΆ, draw a line parallel to π΄π΅ as shown. Extend the parallel lines π΄π΅ and πΆπΈ. Line π΄πΆ is the transversal that intersects the parallel lines.
a. Name the three interior angles of triangle π΄π΅πΆ.
b. Name the straight angle.
c. What kinds of angles are β π΄π΅πΆ and β πΈπΆπ·? What does that mean about their measures?
d. What kinds of angles are β π΅π΄πΆ and β πΈπΆπ΄? What does that mean about their measures?
e. We know that β π΅πΆπ· = β π΅πΆπ΄ + β πΈπΆπ΄ + β πΈπΆπ· = 180Β°. Use substitution to show that the three interior angles of the triangle have a sum of 180Λ.
Lesson 13: Angle Sum of a Triangle Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 13
Exploratory Challenge 2
The figure below shows parallel lines πΏ1 and πΏ2. Let π and π be transversals that intersect πΏ1 at points π΅ and πΆ, respectively, and πΏ2 at point πΉ, as shown. Let π΄ be a point on πΏ1 to the left of π΅, π· be a point on πΏ1 to the right of πΆ, πΊ be a point on πΏ2 to the left of πΉ, and πΈ be a point on πΏ2 to the right of πΉ.
a. Name the triangle in the figure.
b. Name a straight angle that will be useful in proving that the sum of the interior angles of the triangle is 180Λ.
c. Write your proof below.
Lesson 13: Angle Sum of a Triangle Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 13
Problem Set 1. In the diagram below, line π΄π΅ is parallel to line πΆπ·, i.e., πΏπ΄π΅ β₯ πΏπΆπ·. The measure of angle β π΄π΅πΆ = 28Β°, and the
measure of angle β πΈπ·πΆ = 42Β°. Find the measure of angle β πΆπΈπ·. Explain why you are correct by presenting an informal argument that uses the angle sum of a triangle.
2. In the diagram below, line π΄π΅ is parallel to line πΆπ·, i.e., πΏπ΄π΅ β₯ πΏπΆπ·. The measure of angle β π΄π΅πΈ = 38Β° and the measure of angle β πΈπ·πΆ = 16Β°. Find the measure of angle β π΅πΈπ·. Explain why you are correct by presenting an informal argument that uses the angle sum of a triangle. (Hint: find the measure of angle β πΆπΈπ· first, then use that measure to find the measure of angle β π΅πΈπ·.)
Lesson Summary
All triangles have a sum of interior angles equal to 180Λ.
The proof that a triangle has a sum of interior angles equal to 180Λ is dependent upon the knowledge of straight angles and angles relationships of parallel lines cut by a transversal.
Lesson 13: Angle Sum of a Triangle Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 13
3. In the diagram below, line π΄π΅ is parallel to line πΆπ·, i.e., πΏπ΄π΅ β₯ πΏπΆπ·. The measure of angle β π΄π΅πΈ = 56Β°, and the measure of angle β πΈπ·πΆ = 22Β°. Find the measure of angle β π΅πΈπ·. Explain why you are correct by presenting an informal argument that uses the angle sum of a triangle. (Hint: Extend the segment π΅πΈ so that it intersects line πΆπ·.)
4. What is the measure of β π΄πΆπ΅?
5. What is the measure of β πΈπΉπ·?
Lesson 13: Angle Sum of a Triangle Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 13
6. What is the measure of β π»πΌπΊ?
7. What is the measure of β π΄π΅πΆ?
8. Triangle π·πΈπΉ is a right triangle. What is the measure of β πΈπΉπ·?
Lesson 13: Angle Sum of a Triangle Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 13
9. In the diagram below, lines πΏ1 and πΏ2 are parallel. Transversals π and π intersect both lines at the points shown below. Determine the measure of β π½ππΎ. Explain how you know you are correct.
Lesson 14: More on the Angles of a Triangle Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 14
Lesson 14: More on the Angles of a Triangle
Classwork
Exercises 1β4
Use the diagram below to complete Exercises 1β4.
1. Name an exterior angle and the related remote interior angles.
2. Name a second exterior angle and the related remote interior angles.
3. Name a third exterior angle and the related remote interior angles.
4. Show that the measure of an exterior angle is equal to the sum of the related remote interior angles.
Lesson 14: More on the Angles of a Triangle Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 14
Example 1
Find the measure of angle π₯.
Example 2
Find the measure of angle π₯.
Lesson 14: More on the Angles of a Triangle Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 14
Example 3
Find the measure of angle π₯.
Example 4
Find the measure of angle π₯.
Lesson 14: More on the Angles of a Triangle Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 14
Exercises 5β10
5. Find the measure of angle π₯. Present an informal argument showing that your answer is correct.
6. Find the measure of angle π₯. Present an informal argument showing that your answer is correct.
7. Find the measure of angle π₯. Present an informal argument showing that your answer is correct.
Lesson 14: More on the Angles of a Triangle Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 14
8. Find the measure of angle π₯. Present an informal argument showing that your answer is correct.
9. Find the measure of angle π₯. Present an informal argument showing that your answer is correct.
10. Find the measure of angle π₯. Present an informal argument showing that your answer is correct.
Lesson 14: More on the Angles of a Triangle Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 14
Problem Set For each of the problems below, use the diagram to find the missing angle measure. Show your work.
1. Find the measure of angle π₯. Present an informal argument showing that your answer is correct.
2. Find the measure of angle π₯.
Lesson Summary
The sum of the remote interior angles of a triangle is equal to the measure of the exterior angle. For example, β πΆπ΄π΅ + β π΄π΅πΆ = β π΄πΆπΈ.
Lesson 14: More on the Angles of a Triangle Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 14
3. Find the measure of angle π₯. Present an informal argument showing that your answer is correct.
4. Find the measure of angle π₯.
5. Find the measure of angle π₯.
Lesson 14: More on the Angles of a Triangle Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 14
6. Find the measure of angle π₯.
7. Find the measure of angle π₯.
8. Find the measure of angle π₯.
Lesson 14: More on the Angles of a Triangle Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 14
9. Find the measure of angle π₯.
10. Write an equation that would allow you to find the measure of angle π₯. Present an informal argument showing that your answer is correct.
Lesson 15: Informal Proof of the Pythagorean Theorem Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 15
Lesson 15: Informal Proof of the Pythagorean Theorem
Classwork
Example 1
Now that we know what the Pythagorean theorem is, letβs practice using it to find the length of a hypotenuse of a right triangle.
Determine the length of the hypotenuse of the right triangle.
The Pythagorean theorem states that for right triangles π2 + π2 = π2 where π and π are the legs and c is the hypotenuse. Then,
π2 + π2 = π2
62 + 82 = π2
36 + 64 = π2
100 = π2
Since we know that 100 = 102, we can say that the hypotenuse π = 10.
Example 2
Determine the length of the hypotenuse of the right triangle.
Lesson 15: Informal Proof of the Pythagorean Theorem Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 15
Exercises 1β5
For each of the exercises, determine the length of the hypotenuse of the right triangle shown. Note: Figures not drawn to scale. 1.
2.
1.
3.
Lesson 15: Informal Proof of the Pythagorean Theorem Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 15
4.
5.
Lesson 15: Informal Proof of the Pythagorean Theorem Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 15
Problem Set For each of the problems below, determine the length of the hypotenuse of the right triangle shown. Note: Figures not drawn to scale.
1.
2.
Lesson Summary
Given a right triangle π΄π΅πΆ with πΆ being the vertex of the right angle, then the sides π΄πΆ and π΅πΆ are called the legs of βπ΄π΅πΆ and π΄π΅ is called the hypotenuse of βπ΄π΅πΆ.
Take note of the fact that side π is opposite the angle π΄, side π is opposite the angle π΅, and side π is opposite the angle πΆ.
The Pythagorean theorem states that for any right triangle, π2 + π2 = π2.
Lesson 15: Informal Proof of the Pythagorean Theorem Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 15
3.
4.
5.
Lesson 15: Informal Proof of the Pythagorean Theorem Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 15
6.
7.
8.
9.
Lesson 15: Informal Proof of the Pythagorean Theorem Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 15
10.
11.
12.
Lesson 16: Applications of the Pythagorean Theorem Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 16
Lesson 16: Applications of the Pythagorean Theorem
Classwork
Example 1
Given a right triangle with a hypotenuse with length 13 units and a leg with length 5 units, as shown, determine the length of the other leg.
The length of the leg is 12 units.
Exercises
1. Use the Pythagorean theorem to find the missing length of the leg in the right triangle.
52 + π2 = 132 52 β 52 + π2 = 132 β 52
π2 = 132 β 52 π2 = 169 β 25 π2 = 144 π = 12
Lesson 16: Applications of the Pythagorean Theorem Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 16
2. You have a 15-foot ladder and need to reach exactly 9 feet up the wall. How far away from the wall should you place the ladder so that you can reach your desired location?
3. Find the length of the segment π΄π΅.
Lesson 16: Applications of the Pythagorean Theorem Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 16
4. Given a rectangle with dimensions 5 cm and 10 cm, as shown, find the length of the diagonal.
5. A right triangle has a hypotenuse of length 13 in and a leg with length 4 in. What is the length of the other leg?
6. Find the length of π in the right triangle below.
Lesson 16: Applications of the Pythagorean Theorem Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 16
Problem Set 1. Find the length of the segment π΄π΅ shown below.
2. A 20-foot ladder is placed 12 feet from the wall, as shown. How high up the wall will the ladder reach?
Lesson Summary
The Pythagorean theorem can be used to find the unknown length of a leg of a right triangle.
An application of the Pythagorean theorem allows you to calculate the length of a diagonal of a rectangle, the distance between two points on the coordinate plane and the height that a ladder can reach as it leans against a wall.
Lesson 16: Applications of the Pythagorean Theorem Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 16
3. A rectangle has dimensions 6 in by 12 in. What is the length of the diagonal of the rectangle?
Use the Pythagorean theorem to find the missing side lengths for the triangles shown in Problems 4β8.
4. Determine the length of the missing side.
5. Determine the length of the missing side.
6. Determine the length of the missing side.
Lesson 16: Applications of the Pythagorean Theorem Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson 16
7. Determine the length of the missing side.
8. Determine the length of the missing side.