Eureka Math. Geometry Module 1 Student File_B. Additional Student Materials

Transcription

1 A Story of Eureka Math Geometry Module Student File_B Additional Student Materials This file contains: Geo- s Geo- Mid-Module Assessment Geo- End-of-Module Assessment Note that Lesson 5 of this module does not include an. Published by the non-profit GREAT MINDS. Copyright 205 Great Minds. No part of this work may be reproduced, sold, or commercialized, in whole or in part, without written permission from Great Minds. Non-commercial use is licensed pursuant to a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 license; for more information, go to Great Minds and Eureka Math are registered trademarks of Great Minds. Printed in the U.S.A. This book may be purchased from the publisher at eureka-math.org Geo--SFB

2 Packet

3 Lesson Name Date Lesson : Construct an Equilateral Triangle We saw two different scenarios where we used the construction of an equilateral tri angle to hel p determine a needed l ocation (i.e., the fri ends playing catch i n the park and the s itting cats). Can you think of another scenario where the construction of an equilateral triangle might be useful? Articulate how you would find the needed location using an equilateral triangle. Lesson : Construct an Equilateral Triangle

4 Lesson 2 Name Date Lesson 2: Construct an Equilateral Triangle is shown below. Is it an equilateral triangle? Justify your response. Lesson 2: Construct an Equilateral Triangle

5 Lesson 3 Name Date Lesson 3: Copy and Bisect an Angle Later that day, Ji mmy and Joey were working together to build a kite wi th sticks, newspapers, tape, and string. After they fastened the sti cks together i n the overall shape of the kite, Ji mmy l ooked at the position of the sticks and said that each of the four corners of the kite is bisected; Joey said that they would only be able to bisect the top and bottom angles of the kite. Who i s correct? Explain. Lesson 3: Copy and Bisect an Angle

6 Lesson 4 Name Date Lesson 4: Construct a Perpendicular Bisector Divide the following segment into four segments of equal l ength. Lesson 4: Construct a Perpendicular Bisector

7 Lesson 6 Name Date Lesson 6: Solve for Unknown Angles Angles and Lines at a Point Use the following diagram to answer the questions below:. a. Name an angle supplementary to, and provide the reason for your calculation. b. Name an angle complementary to, and provide the reason for your calculation. 2. If = 38, what i s the measure of each of the following angles? Provide reasons for your calculations. a. b. c. Lesson 6: Solve for Unknown Angles Angles and Lines at a Point

8 Lesson 7 Name Date Lesson 7: Solving for Unknown Angles Transversals Determine the value of each variable. = = = Lesson 7: Solve for Unknown Angles Transversals

9 Lesson 8 Name Date Lesson 8: Solve for Unknown Angles Angles in a Triangle Find the value of and. = = Lesson 8: Solve for Unknown Angles Angles in a Triangle

10 Lesson 9 Name Date Lesson 9: Unknown Angle Proofs Writing Proofs In the diagram to the ri ght, prove that the sum of the l abeled angles is 80. Lesson 9: Unknown Angle Proofs Writing Proofs

11 Lesson 0 Name Date Lesson 0: Unknown Angle Proofs Proofs with Constructions Write a proof for each question.. In the figure to the ri ght,. Prove that =. 2. Prove =. Lesson 0: Unknown Angle Proofs Proofs with Constructions

12 Lesson Name Date Lesson : Unknown Angle Proofs Proofs of Known Facts In the diagram to the ri ght, prove that + =80. Lesson : Unknown Angle Proofs Proofs of Known Facts

13 Lesson 2 Name Date Lesson 2: Transformations The Next Level How are transformations and functions related? Provide a specific example to support your reasoning. Lesson 2: Transformations The Next Level

14 Lesson 3 Name Date Lesson 3: Rotations Find the center of rotation and the angle of rotation for the transformation below that carries onto. Lesson 3: Rotations

15 Lesson 4 Name Date Lesson 4: Reflections. Construct the line of reflection for the figures. 2. Reflect the given pre-image across the line of reflection provided. Lesson 4: Reflections

16 Lesson 5 Name Date Lesson 5: Rotations, Reflections, and Symmetry What i s the rel ationship between a rotation and a reflection? Sketch a diagram that supports your explanation. Lesson 5: Rotations, Reflections, and Symmetry

17 Lesson 6 Name Date Lesson 6: Translations Translate the figure one unit down and three units right. Draw the vector that defines the translation. Lesson 6: Translations

18 Lesson 7 Name Date Lesson 7: Characterize Points on a Perpendicular Bisector Using your understanding of rigid motions, explain why any point on the perpendicular bisector is equidistant from any pair of pre-image and image points. Use your construction tools to create a figure that supports your explanation. Lesson 7: Characterize Points on a Perpendicular Bisector

19 Lesson 8 Name Date Lesson 8: Looking More Carefully at Parallel Lines. Construct a line through the point bel ow that i s parallel to the l i ne by rotating by 80 (using the steps outlined i n Example 2). 2. Why i s the parallel l ine you constructed the only l ine that contains and is parallel to? Lesson 8: Looking More Carefully at Parallel Lines

20 Lesson 9 Name Date Lesson 9: Construct and Apply a Sequence of Rigid Motions Assume that the following figures are drawn to scale. Use your understanding of congruence to explain why square and rhombus are not congruent. Lesson 9: Construct and Apply a Sequence of Rigid Motions

21 Lesson 20 Name Date Lesson 20: Applications of Congruence in Terms of Rigid Motions. What i s a correspondence? Why does a congruence naturally yield a correspondence? 2. Each side of is twice the length of each side of. Fil l i n the blanks bel ow so that each rel ationship between lengths of sides is true. 2 = 2 = 2 = Lesson 20: Applications of Congruence in Terms of Rigid Motions

22 Lesson 2 Name Date Lesson 2: Correspondence and Transformations Complete the table based on the series of rigid moti ons performed on below. Sequence of Rigid Motions (2) Composition in Function Notation Sequence of Corresponding Sides Sequence of Corresponding Angles Triangle Congruence Statement Lesson 2: Correspondence and Transformations

23 Lesson 22 Name Date Lesson 22: Congruence Criteria for Triangles SAS If two tri angles satisfy the SAS cri teria, describe the ri gid motion(s) that would map one onto the other in the following cases.. The two triangles share a single common vertex. 2. The two triangles are distinct from each other. 3. The two triangles share a common side. Lesson 22: Congruence Criteria for Triangles SAS

24 Lesson 23 Name Date Lesson 23: Base Angles of Isosceles Triangles For each of the following, if the given congruence exists, name the isosceles triangle and the pair of congruent angles for the triangle based on the image above Lesson 23: Base Angles of Isosceles Triangles

25 Lesson 24 Name Date Lesson 24: Congruence Criteria for Triangles ASA and SSS Based on the information provided, determine whether a congruence exists between triangles. If a congruence exists between triangles or if multiple congruencies exist, state the congruencies and the criteria used to determine them. Given: =, is the midpoint of Lesson 24: Congruence Criteria for Triangles ASA and SSS

26 Lesson 25 Name Date Lesson 25: Congruence Criteria for Triangles AAS and HL. Sketch an example of two tri angles that meet the AAA cri teria but are not congruent. 2. Sketch an example of two triangles that meet the SSA criteria that are not congruent. Lesson 25: Congruence Criteria for Triangles AAS and HL

27 Lesson 26 Name Date Lesson 26: Triangle Congruency Proofs Identi fy the two tri angle congruence criteria that do NOT guarantee congruence. Explain why they do not guarantee congruence, and provide illustrations that support your reasoning. Lesson 26: Triangle Congruency Proofs

28 Lesson 27 Name Date Lesson 27: Triangle Congruency Proofs Given: Prove: is the midpoint of, Lesson 27: Triangle Congruency Proofs

29 Lesson 28 Name Date Lesson 28: Properties of Parallelograms Given: Prove: Equilateral parallelogram (i.e., a rhombus) with diagonals and Diagonals intersect perpendicularly. Lesson 28: Properties of Parallelograms

30 Lesson 29 Name Date Lesson 29: Special Lines in Triangles Use the properties of midsegments to solve for the unknown value in each question.. and are the mi dpoints of and, respectively. What is the perimeter of? 2. What is the perimeter of? Lesson 29: Special Lines in Triangles

31 Lesson 30 Name Date Lesson 30: Special Lines in Triangles,, and are all medians of, and is the centroid. =24, =0, =7. Find,,, and. Lesson 30: Special Lines in Triangles

32 Lesson 3 Name Date Lesson 3: Construct a Square and a Nine-Point Circle Construct a square and a square so that contains and contains. Lesson 3: Construct a Square and a Nine-Point Circle

33 Lesson 32 Name Date Lesson 32: Construct a Nine-Point Circle Construct a nine-point circle, and then i nscribe a square i n the ci rcle (so that the verti ces of the square are on the ci rcle). Lesson 32: Construct a Nine-Point Circle

34 Lesson 33 Name Date Lesson 33: Review of the Assumptions. Which assumption(s) must be used to prove that vertical angles are congruent? 2. If two l i nes are cut by a transversal such that corresponding angles are NOT congruent, what must be true? Justify your response. Lesson 33: Review of the Assumptions

35 Lesson 34 Name Date Lesson 34: Review of the Assumptions The inner parallelogram in the figure is formed from the midsegments of the four triangles created by the outer parallelogram s diagonals. The l engths of the smaller and l arger midsegments are as i ndicated. If the perimeter of the outer parallelogram is 40, find the val ue of. Lesson 34: Review of the Assumptions

36 Assessment Packet

37 Mid-Module Assessment Task Name Date. State precise definitions of angle, circle, perpendicular, parallel, and line segment based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Angle: Circle: Perpendicular: Parallel: Line segment: Module : Congruence, Proof, and Constructions

38 Mid-Module Assessment Task 2. A rigid motion,, of the plane takes a point,, as input and gives as output (i.e., ( ) = ). Similarly, ( ) = for input point and output point. Jerry claims that knowing nothing else about, we can be sure that because rigid motions preserve distance. a. Show that Jerry s claim is incorrect by giving a counterexample (hint: a counterexample would be a specific rigid motion and four points,,, and in the plane such that the motion takes to and to, yet ). b. There is a type of rigid motion for which Jerry s claim is always true. Which type below is it? Rotation Reflection Translation c. Suppose Jerry claimed that. Would this be true for any rigid motion that satisfies the conditions described in the first paragraph? Why or why not? Module : Congruence, Proof, and Constructions 2

39 Mid-Module Assessment Task 3. a. In the diagram below, is a line, is a point on the line, and is a point not on the line. is the midpoint of. Show how to create a line parallel to that passes through by using a rotation about. b. Suppose that four lines in a given plane,,,, and are given, with the conditions (also given) that,, and is neither parallel nor perpendicular to. i. Sketch (freehand) a diagram of,,, and to illustrate the given conditions. ii. In any diagram that illustrates the given conditions, how many distinct angles are formed? Count only angles that measure less than 80, and count two angles as the same only if they have the same vertex and the same edges. Among these angles, how many different angle measures are formed? Justify your answer. Module : Congruence, Proof, and Constructions 3

40 Mid-Module Assessment Task 4. In the figure below, there is a reflection that transforms to. Use a straightedge and compass to construct the line of reflection, and list the steps of the construction. Module : Congruence, Proof, and Constructions 4

41 Mid-Module Assessment Task 5. Precisely define each of the three rigid motion transformations identified. a. ( ) b. ( ) c., ( ) Module : Congruence, Proof, and Constructions 5

42 Mid-Module Assessment Task 6. Given in the figure below, line is the perpendicular bisector of and of. a. Show using rigid motions. b. Show. c. Show. Module : Congruence, Proof, and Constructions 6

43 End-of-Module Assessment Task Name Date. Each of the illustrations on the next page shows in black a plane figure consisting of the letters F, R, E, and D evenly spaced and arranged in a row. In each illustration, an alteration of the black figure is shown in gray. In some of the illustrations, the gray figure is obtained from the black figure by a geometric transformation consisting of a single rotation. In others, this is not the case. a. Which illustrations show a single rotation? b. Some of the illustrations are not rotations or even a sequence of rigid transformations. Select one such illustration, and use it to explain why it is not a sequence of rigid transformations. Module : Congruence, Proof, and Constructions

44 End-of-Module Assessment Task Module : Congruence, Proof, and Constructions 2

45 End-of-Module Assessment Task 2. In the figure below, bisects, =, = 90, and = 42. Find the measure of. Module : Congruence, Proof, and Constructions 3

46 End-of-Module Assessment Task 3. In the figure below, is the angle bisector of. and are straight lines, and. Prove that =. Module : Congruence, Proof, and Constructions 4

47 End-of-Module Assessment Task 4. and, in the figure below are such that,, and. a. Which criteria for triangle congruence (ASA, SAS, SSS) implies that? b. Describe a sequence of rigid transformations that shows. Module : Congruence, Proof, and Constructions 5

48 End-of-Module Assessment Task 5. a. Construct a square with side. List the steps of the construction. Module : Congruence, Proof, and Constructions 6

49 End-of-Module Assessment Task b. Three rigid motions are to be performed on square. The first rigid motion is the reflection through. The second rigid motion is a 90 clockwise rotation around the center of the square. Describe the third rigid motion that will ultimately map back to its original position. Label the image of each rigid motion,,, and in the blanks provided. Rigid Motion Description: Reflection through Rigid Motion 2 Description: 90 clockwise rotation around the center of the square. Rigid Motion 3 Description: Module : Congruence, Proof, and Constructions 7

50 End-of-Module Assessment Task 6. Suppose that is a parallelogram and that and are the midpoints of and, respectively. Prove that is a parallelogram. Module : Congruence, Proof, and Constructions 8