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Lesson 1 NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 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). ο§ ο§ ο§ ο§ 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. ο§ 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. Problem Set 1. Using as much of the new vocabulary as you can, try to describe what you see in the diagram below. F(B) B F(A) A 2. Describe, intuitively, what kind of transformation will be required to move Figure A on the left to its image on the right. Figure A Image of A Lesson 1: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Why Move Things Around? 9/17/13 S.3 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 2 NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson Summary Translation occurs along a given vector: ο§ 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. ο§ The length of a vector is, by definition, the length of its underlying segment. ο§ 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. Problem Set 1. Translate the plane containing Figure A along οΏ½οΏ½οΏ½οΏ½οΏ½β π΄π΅ . 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 οΏ½οΏ½οΏ½οΏ½οΏ½β π΅π΄ 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 2: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Definition of Translation and Three Basic Properties 9/17/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. S.6 Lesson 2 NYS COMMON CORE MATHEMATICS CURRICULUM 8β’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 οΏ½οΏ½οΏ½οΏ½οΏ½β π΄π΅ and label the image π·β². What do you notice about the line containing vector οΏ½οΏ½οΏ½οΏ½οΏ½β π΄π΅ , and the line containing points π· and π·β²? (Hint: Will the lines ever intersect?) 8. Translate point πΈ along vector οΏ½οΏ½οΏ½οΏ½οΏ½β π΄π΅ and label the image πΈβ². What do you notice about the line containing vector οΏ½οΏ½οΏ½οΏ½οΏ½β π΄π΅ , and the line containing points πΈ and πΈβ²? Lesson 2: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Definition of Translation and Three Basic Properties 9/17/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. S.7 Lesson 3 NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson Summary ο§ Two lines are said to be parallel if they do not intersect. ο§ Translations map parallel lines to parallel lines. ο§ 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. Problem Set 1. Translate β πππ, point π΄, point π΅, and rectangle π»πΌπ½πΎ along vector οΏ½οΏ½οΏ½οΏ½οΏ½β πΈπΉ 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. 4. 5. οΏ½οΏ½οΏ½οΏ½οΏ½β ? Connect π΅ to π΅β². What do you know about the line formed by π΅π΅β² and the line containing the vector πΈπΉ οΏ½οΏ½οΏ½οΏ½οΏ½β ? Connect π΄ to π΄β². What do you know about the line formed by π΄π΄β² and the line containing the vector πΈπΉ Given that figure π»πΌπ½πΎ is a rectangle, what do you know about lines π»πΌ and π½πΎ and their translated images? Explain. Lesson 3: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Translating Lines 9/17/13 S.11 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 4 NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson Summary ο§ A reflection is another type of basic rigid motion. ο§ Reflections occur across lines. The line that you reflect across is called the line of reflection. ο§ When a point, π, is joined to its reflection, πβ², the line of reflection bisects the segment, ππβ² . Problem Set 1. In the picture below, β π·πΈπΉ = 56°, β π΄πΆπ΅ = 114°, 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 4: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Definition of Reflection and Basic Properties 9/17/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. S.16 Lesson 4 NYS COMMON CORE MATHEMATICS CURRICULUM 2. 3. 4. 5. 6. 7. 8β’2 What is the size of π ππππππ‘πππ(β π·πΈπΉ)? Explain. What is the length of π ππππππ‘πππ(π½πΎ)? Explain. What is the size of π ππππππ‘πππ(β π΄πΆπ΅)? What is the length of π ππππππ‘πππ(π΄π΅)? Two figures in the picture were not moved under the reflection. Name the two figures and explain why they were not moved. 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 4: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Definition of Reflection and Basic Properties 9/17/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. S.17 Lesson 5 NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 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. Problem Set 1. Let there be a rotation by β 90Λ around the center π. Lesson 5: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Definition of Rotation and Basic Properties 9/17/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. S.23 Lesson 5 NYS COMMON CORE MATHEMATICS CURRICULUM 2. 8β’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 5: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Definition of Rotation and Basic Properties 9/17/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. S.24 Lesson 6 NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson Summary ο§ ο§ 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. Problem Set Use the following diagram for problems 1β5. Use your transparency, as needed. 1. 2. 3. 4. 5. Looking only at segment π΅πΆ, is it possible that a 180Λ rotation would map π΅πΆ onto π΅β²πΆβ²? Why or why not? 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? Looking only at segment π΄πΆ, is it possible that a 180Λ rotation would map π΄πΆ onto π΄β²πΆβ²? Why or why not? 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? 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 6: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Rotations of 180 Degrees 9/18/13 S.31 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM 6. Lesson 6 8β’2 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΄β² π΅β² = 1, and π΅πΆ = π΅β² πΆ β² = 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 6: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Rotations of 180 Degrees 9/18/13 S.32 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 7 NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson Summary ο§ Translating a figure along one vector then translating its image along another vector is an example of a sequence of transformations. ο§ A sequence of translations enjoys the same properties as a single translation. Specifically, the figuresβ lengths and degrees of angles are preserved. ο§ 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. Problem Set 1. Sequence translations of Parallelogram π΄π΅πΆπ· (a quadrilateral in which both pairs of opposite sides are parallel) πΉπΈ . Label the translated images. along vectors οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½β π»πΊ and οΏ½οΏ½οΏ½οΏ½οΏ½β 2. What do you know about π΄π· and π΅πΆ compared with π΄β²π·β² and π΅β²πΆβ²? Explain. 3. 4. Are π΄β²π΅β² and π΄β²β² π΅β²β² equal in length? How do you know? Translate the curved shape π΄π΅πΆ along the given vector. Label the image. Lesson 7: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Sequencing Translations 9/18/13 S.38 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM 5. Lesson 7 8β’2 What vector would map the shape π΄β²π΅β²πΆβ² back onto π΄π΅πΆ? Lesson 7: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Sequencing Translations 9/18/13 S.39 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 8 8β’2 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. Problem Set 1. Let there be a reflection across line πΏ and let there be a translation along vector οΏ½οΏ½οΏ½οΏ½οΏ½β π΄π΅ 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 8: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Sequencing Reflections and Translations 9/18/13 S.44 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 9 NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 Lesson Summary ο§ 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. ο§ 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. ο§ 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. Problem Set 1. Refer to the figure below. a. b. c. 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. What is the size of β π΄π΅πΆ and how does it compare to the size of β π΄β² π΅β² πΆ β² ? Explain. What is the length of segment DE and how does it compare to the length of segment π·β²πΈβ²? Explain. Lesson 9: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Sequencing Rotations 9/18/13 S.50 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 9 NYS COMMON CORE MATHEMATICS CURRICULUM 2. Refer to the figure given below. a. b. 3. 8β’2 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. Describe the sequence of rigid motions that would map β³ π΄π΅πΆ onto β³ π΄β² π΅β² πΆ β² . 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 9: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Sequencing Rotations 9/18/13 S.51 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 11 NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 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 . 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 11: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Definition of Congruence and Some Basic Properties 9/18/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. S.61 Lesson 11 NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 3. a. β² π΄β² . οΏ½οΏ½οΏ½οΏ½οΏ½β and οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½β οΏ½οΏ½οΏ½οΏ½οΏ½β to π οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½β Given two rays, ππ΄ πβ² π΄β² , describe the congruence that maps ππ΄ b. οΏ½οΏ½οΏ½οΏ½οΏ½β. Describe the congruence that maps οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½β πβ² π΄β² to ππ΄ Lesson 11: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Definition of Congruence and Some Basic Properties 9/18/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. S.62 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 12 8β’2 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. Problem Set Use the diagram below to do Problems 1β6. Lesson 12: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Angles Associated with Parallel Lines 9/18/13 S.65 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 12 NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 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. 5. 6. 7. 8. 9. Assuming πΏ1 β₯ πΏ2 if the measure of β 4 is 73Λ, what is the measure of β 8? How do you know? Assuming πΏ1 β₯ πΏ2 , if the measure of β 3 is 107Λ degrees, what is the measure of β 6? How do you know? Assuming πΏ1 β₯ πΏ2 , if the measure of β 2 is 107Λ, what is the measure of β 7? How do you know? Would your answers to Problems 4β6 be the same if you had not been informed that πΏ1 β₯ πΏ2 ? Why or why not? Use an informal argument to describe why β 1 and β 5 are equal in measure if πΏ1 β₯ πΏ2 . 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 12: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Angles Associated with Parallel Lines 9/18/13 S.66 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 13 8β’2 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. 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 = 28°, and the measure of angle β πΈπ·πΆ = 42°. 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 = 38° and the measure of angle β πΈπ·πΆ = 16°. 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 13: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Angle Sum of a Triangle 9/18/13 S.70 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 13 8β’2 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 = 56°, and the measure of angle β πΈπ·πΆ = 22°. 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: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Angle Sum of a Triangle 9/18/13 S.71 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM 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: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Lesson 13 8β’2 Angle Sum of a Triangle 9/18/13 S.72 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM 9. Lesson 13 8β’2 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 13: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Angle Sum of a Triangle 9/18/13 S.73 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 14 8β’2 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. 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 14: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org More on the Angles of a Triangle 9/18/13 S.79 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM 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: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org 8β’2 More on the Angles of a Triangle 9/18/13 S.80 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM 6. Find the measure of angle π₯. 7. Find the measure of angle π₯. 8. Find the measure of angle π₯. Lesson 14: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Lesson 14 8β’2 More on the Angles of a Triangle 9/18/13 S.81 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM 9. Lesson 14 8β’2 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 14: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org More on the Angles of a Triangle 9/18/13 S.82 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 15 NYS COMMON CORE MATHEMATICS CURRICULUM 8β’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 + π 2 = π 2 . 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 15: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Informal Proof of the Pythagorean Theorem 9/18/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. S.86 Lesson 15 NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 3. 4. 5. Lesson 15: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Informal Proof of the Pythagorean Theorem 9/18/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. S.87 Lesson 15 NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 6. 7. 8. 9. Lesson 15: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Informal Proof of the Pythagorean Theorem 9/18/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. S.88 Lesson 15 NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 10. 11. 12. Lesson 15: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Informal Proof of the Pythagorean Theorem 9/18/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. S.89 Lesson 16 NYS COMMON CORE MATHEMATICS CURRICULUM 8β’2 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. 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 16: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Applications of the Pythagorean Theorem 9/18/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. S.93 Lesson 16 NYS COMMON CORE MATHEMATICS CURRICULUM 3. 8β’2 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: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org Applications of the Pythagorean Theorem 9/18/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. S.94 Lesson 16 NYS COMMON CORE MATHEMATICS CURRICULUM 7. Determine the length of the missing side. 8. Determine the length of the missing side. Lesson 16: Date: © 2013 Common Core, Inc. Some rights reserved. commoncore.org 8β’2 Applications of the Pythagorean Theorem 9/18/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. S.95