New York State Common Core

Mathematics Curriculum
GEOMETRY • MODULE 1

Table of Contents1

Congruence, Proof, and Constructions
Module Overview .................................................................................................................................................. 3 Topic A: Basic Constructions (G-CO.1, G-CO.12, G-CO.13).................................................................................... 7 Lesson 1: Construct an Equilateral Triangle ............................................................................................. 8 Lesson 2: Construct an Equilateral Triangle II ........................................................................................ 16 Lesson 3: Copy and Bisect an Angle........................................................................................................ 21 Lesson 4: Construct a Perpendicular Bisector ........................................................................................ 30 Lesson 5: Points of Concurrencies .......................................................................................................... 37 Topic B: Unknown Angles (G-CO.9) ..................................................................................................................... 43 Lesson 6: Solve for Unknown Angles—Angles and Lines at a Point ....................................................... 44 Lesson 7: Solve for Unknown Angles—Transversals .............................................................................. 52 Lesson 8: Solve for Unknown Angles—Angles in a Triangle ................................................................... 60 Lesson 9: Unknown Angle Proofs—Writing Proofs ................................................................................ 66 Lesson 10: Unknown Angle Proofs—Proofs with Constructions ............................................................ 72 Lesson 11: Unknown Angle Proofs—Proofs of Known Facts.................................................................. 78 Topic C: Transformations/Rigid Motions (G-CO.2, G-CO.3, G-CO.4, G-CO.5, G-CO.6, G-CO.7, G-CO.12)........... 84 Lesson 12: Transformations—The Next Level ........................................................................................ 86 Lesson 13: Rotations ............................................................................................................................... 95 Lesson 14: Reflections .......................................................................................................................... 104 Lesson 15: Rotations, Reflections, and Symmetry ............................................................................... 111 Lesson 16: Translations ........................................................................................................................ 117 Lesson 17: Characterize Points on a Perpendicular Bisector................................................................ 125 Lesson 18: Looking More Carefully at Parallel Lines ............................................................................ 131 Lesson 19: Construct and Apply a Sequence of Rigid Motions ............................................................ 141 Lesson 20: Applications of Congruence in Terms of Rigid Motions ..................................................... 146
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Each lesson is ONE day and ONE day is considered a 45 minute period.

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NYS COMMON CORE MATHEMATICS CURRICULUM

Module Overview

M1

GEOMETRY

Lesson 21: Correspondence and Transformations ............................................................................... 153 Mid-Module Assessment and Rubric ................................................................................................................ 158 Topics A through C (assessment 1 day, return 1 day, remediation or further applications 2 days) Topic D: Congruence (G-CO.7, G-CO.8) ............................................................................................................. 172 Lesson 22: Congruence Criteria for Triangles—SAS ............................................................................. 173 Lesson 23: Base Angles of Isosceles Triangles ...................................................................................... 181 Lesson 24: Congruence Criteria for Triangles—ASA and SSS ............................................................... 188 Lesson 25: Congruence Criteria for Triangles—SAA and HL ................................................................. 195 Lesson 26: Triangle Congruency Proofs—Part I.................................................................................... 202 Lesson 27: Triangle Congruency Proofs—Part II................................................................................... 206 Topic E: Proving Properties of Geometric Figures (G-CO.9, G-CO.10, G-CO.11) ............................................... 211 Lesson 28: Properties of Parallelograms .............................................................................................. 212 Lessons 29-30: Special Lines in Triangles.............................................................................................. 222 Topic F: Advanced Constructions (G-CO.13) ..................................................................................................... 232 Lesson 31: Construct a Square and a Nine-Point Circle........................................................................ 233 Lesson 32: Construct a Nine-Point Circle.............................................................................................. 238 Topic G: Axiomatic Systems (G-CO.1, G-CO.2, G-CO.3, G-CO.4, G-CO.5, G-CO.6, G-CO.7, G-CO.8, G-CO.9, G-CO.10, G-CO.11, G-CO.12, G-CO.13) .................................................................................. 242 Lessons 33-34: Review of the Assumptions ......................................................................................... 244 End-of-Module Assessment and Rubric ............................................................................................................ 257 Topics A through G (assessment 1 day, return 1 day, remediation or further applications 3 days)

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NYS COMMON CORE MATHEMATICS CURRICULUM

Module Overview

M1

GEOMETRY

Geometry • Module 1

Congruence, Proof, and Constructions
OVERVIEW
Module 1 embodies critical changes in Geometry as outlined by the Common Core. The heart of the module is the study of transformations and the role transformations play in defining congruence. Students begin this module with Topic A, Constructions. Major constructions include an equilateral triangle, an angle bisector, and a perpendicular bisector. Students synthesize their knowledge of geometric terms with the use of new tools and simultaneously practice precise use of language and efficient communication when they write the steps that accompany each construction (G.CO.1). Constructions segue into Topic B, Unknown Angles, which consists of unknown angle problems and proofs. These exercises consolidate students’ prior body of geometric facts and prime students’ reasoning abilities as they begin to justify each step for a solution to a problem. Students began the proof writing process in Grade 8 when they developed informal arguments to establish select geometric facts (8.G.5). Topic C, Transformations, builds on students’ intuitive understanding developed in Grade 8. With the help of manipulatives, students observed how reflections, translations, and rotations behave individually and in sequence (8.G.1, 8.G.2). In Grade 10, this experience is formalized by clear definitions (G.CO.4) and more indepth exploration (G.CO.3, G.CO.5). The concrete establishment of rigid motions also allows proofs of facts formerly accepted to be true (G.CO.9). Similarly, students’ Grade 8 concept of congruence transitions from a hands-on understanding (8.G.2) to a precise, formally notated understanding of congruence (G.CO.6). With a solid understanding of how transformations form the basis of congruence, students next examine triangle congruence criteria. Part of this examination includes the use of rigid motions to prove how triangle congruence criteria such as SAS actually work (G.CO.7, G.CO.8). In Topic D, Proving Properties of Geometric Figures, students use what they have learned in Topics A through C to prove properties—those that have been accepted as true and those that are new—of parallelograms and triangles (G.CO.10, G.CO.11). The module closes with a return to constructions in Topic E (G.CO.13), followed by a review that of the module that highlights how geometric assumptions underpin the facts established thereafter (Topic F).

Focus Standards
Experiment with transformations in the plane.
G-CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

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NYS COMMON CORE MATHEMATICS CURRICULUM

Module Overview

M1

GEOMETRY

G-CO.2

Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

G-CO.3 G-CO.4 G-CO.5

Understand congruence in terms of rigid motions.
G-CO.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

G-CO.7

G-CO.8

Prove geometric theorems.
G-CO.9 Prove 2 theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove2 theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Prove2 theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

G-CO.10

G-CO.11

2

Prove and apply (in preparation for Regents Exams).

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NYS COMMON CORE MATHEMATICS CURRICULUM

Module Overview

M1

GEOMETRY

Make geometric constructions.
G-CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

G-CO.13

Foundational Standards
Understand congruence and similarity using physical models, transparencies, or geometry software.
8.G.1 Verify experimentally the properties of rotations, reflections, and translations: a. b. c. 8.G.2 Lines are taken to lines, and line segments to line segments of the same length. Angles are taken to angles of the same measure. Parallel lines are taken to parallel lines.

Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them. Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.

8.G.3 8.G.5

Focus Standards for Mathematical Practice
MP.3 Construct viable arguments and critique the reasoning of others. Students articulate steps needed to construct geometric figures, using relevant vocabulary. Students develop and justify conclusions about unknown angles and defend their arguments with geometric reasons. Model with mathematics. Students apply geometric constructions and knowledge of rigid motions to solve problems arising with issues of design or location of facilities. Use appropriate tools strategically. Students consider and select from a variety of tools in constructing geometric diagrams, including (but not limited to) technological tools.

MP.4 MP.5

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NYS COMMON CORE MATHEMATICS CURRICULUM

Module Overview

M1

GEOMETRY

MP.6

Attend to precision. Students precisely define the various rigid motions. Students demonstrate polygon congruence, parallel status, and perpendicular status via formal and informal proofs. In addition, students will clearly and precisely articulate steps in proofs and constructions throughout the module.

Terminology
New or Recently Introduced Terms
 Isometry

Familiar Terms and Symbols 3
     Transformation Translation Rotation Reflection Congruence

Suggested Tools and Representations
   Compass and straightedge Geometer’s Sketchpad or Geogebra Software Patty paper

Assessment Summary
Assessment Type
Mid-Module Assessment Task End-of-Module Assessment Task

Administered Format
After Topic C Constructed response with rubric

Standards Addressed
G-CO.1, G-CO.2, G-CO.4, G-CO.5, G-CO.6, G-CO.9, G-CO.12 G-CO.2, G-CO.3, G-CO.7, G-CO.8, G-CO.10, GCO.11, G-CO.13

After Topic G

Constructed response with rubric

3

These are terms and symbols students have seen previously.

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New York State Common Core

Mathematics Curriculum
GEOMETRY • MODULE 1

Topic A:

Basic Constructions
G-CO.1, G-CO.12, G-CO.13
Focus Standard: G-CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

G-CO.12

G-CO.13 Instructional Days: 5

Lesson 1: Construct an Equilateral Triangle (M)1 Lesson 2: Construct an Equilateral Triangle II (E) Lesson 3: Copy and Bisect an Angle (M) Lesson 4: Construct a Perpendicular Bisector (M) Lesson 5: Points of Concurrencies (E)

The first module of Geometry incorporates and formalizes geometric concepts presented in all the different grade levels up to tenth grade. Topic A brings the relatively unfamiliar concept of construction to life by building upon ideas students are familiar with, such as the constant length of the radius within a circle. While the figures that are being constructed may not be novel, the process of using tools to create the figures is certainly new. Students use construction tools, such as a compass, straightedge, and patty paper, to create constructions of varying difficulty, including equilateral triangles, perpendicular bisectors, and angle bisectors. The constructions are embedded in models that require students to make sense of their space in addition to understanding how to find an appropriate solution with their tools. Students will also discover the critical need for precise language when they articulate the steps necessary for each construction. The figures covered throughout the topic provide a bridge to solving, then proving, unknown angle problems.

1

Lesson Structure Key: P-Problem Set Lesson, M-Modeling Cycle Lesson, E-Exploration Lesson, S-Socratic Lesson

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NYS COMMON CORE MATHEMATICS CURRICULUM

Lesson 1

M1

GEOMETRY

Lesson 1: Construct an Equilateral Triangle
Student Outcomes
  Students learn to construct an equilateral triangle. Students communicate mathematic ideas effectively and efficiently.

Lesson Notes
Most students will have done little more than draw a circle with a compass upon entering 10 th grade. The first few lessons on constructions will be a topic where students truly acquire a whole new set of skills. This lesson begins with a brief Opening Exercise, which requires peer-to-peer conversation, and attention to vocabulary. Ensure students understand that, even though the vocabulary terms may be familiar, students should pay careful attention to the precision of each definition. For students to develop logical reasoning in geometry, they have to manipulate very exact language, beginning with definitions. Students explore various phrasings of definitions. The teacher guides the discussion until students arrive at a formulation of the standard definition. The purpose of the discussion is to understand why the definition has the form that it does. As part of the discussion, students should be able to test the strength of any definition by looking for possible counterexamples. Sitting Cats, the main activity, provides a backdrop to constructing the equilateral triangle. Though students may visually understand where the position of the third cat should be, they will spend time discovering how to use their compass to establish the exact location. (The cat, obviously, will be in a position that approximates the third vertex. The point constructed is the optimal position of the cat- if cats were points and were perfect in their choice of place to sleep.) Students should work without assistance for some portion of the 15 minutes allotted. As students begin to successfully complete the task, elicit what about the use of the compass makes this construction possible. In the last segment of class, the teacher should lead students through Euclid’s Proposition 1 of Book 1 (Elements 1:1). Have students annotate the text as they read, noting how labeling is used to direct instructions. After reading through the document, direct students to write in their own words the steps they took to construct an equilateral triangle. As part of the broader goal of teaching students to communicate precisely and effectively in geometry, emphasize the need for clear instruction, for labeling in their diagram and reference to labeling in the steps, and for coherent use of relevant vocabulary. Students should begin the process in class together but should complete the assignment for Homework.

Classwork
Opening Exercise (10 minutes)
Students should brainstorm ideas in pairs. Students may think of the use of counting footsteps, rope, or measuring tape to make the distances between friends precise.

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NYS COMMON CORE MATHEMATICS CURRICULUM

Lesson 1

M1

GEOMETRY

Opening Exercise Joe and Marty are in the park playing catch. Tony joins them, and the boys want to stand so that the distance between any two of them is the same. Where do they stand? How do they figure this out precisely? What tool or tools could they use?

Fill in the blanks below as each term is discussed: The _______ between points ������ and ������ is the set consisting of ������, ������, and all points ⃡ on the line ������������ between ������ and ������. A segment from the center of a circle to a point on the circle. Given a point ������ in the plane and a number ������ > ������, the _______ with center ������ and radius ������ is the set of all points in the plane that are distance ������ from the point ������.

1. 2. 3.

Segment Radius Circle

Example 1 (10 minutes): Sitting Cats
Students explore how to construct an equilateral triangle using a compass.
Example 1: Sitting Cats You will need a compass and a straightedge. Margie has three cats. She has heard that cats in a room position themselves at equal distances from one another and wants to test that theory. Margie notices that Simon, her tabby cat, is in the center of her bed (at S), while JoJo, her Siamese, is lying on her desk chair (at J). If the theory is true, where will she find Mack, her calico cat? Use the scale drawing of Margie’s room shown below, together with (only) a compass and straightedge. Place an M where Mack will be if the theory is true.

MP.5

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NYS COMMON CORE MATHEMATICS CURRICULUM

Lesson 1

M1

GEOMETRY

Example 2 (12 minutes): Euclid, Proposition 1
Students examine Euclid’s solution of how to construct an equilateral triangle. Lead students through this excerpt and have them annotate the text as they read it. The goal is for students to form a rough set of steps that outlines the construction of the equilateral triangle. Once a first attempt of the steps is made, review them as if you are using them as a step-by-step guide. Ask the class if the steps need refinement. This is to build to the homework question, which asks students to write a clear and succinct set of instructions for the construction of the equilateral triangle.
Example 2: Euclid, Proposition 1 Let’s see how Euclid approached this problem. Look his first proposition, and compare his steps with yours.

In this margin, compare your steps with Euclid’s.

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NYS COMMON CORE MATHEMATICS CURRICULUM

Lesson 1

M1

GEOMETRY

Geometry Assumptions (7 minutes)
Geometry Assumptions In geometry, as in most fields, there are specific facts and definitions that we assume to be true. In any logical system, it helps to identify these assumptions as early as possible since the correctness of any proof we offer hinges upon the truth of our assumptions. For example, in Proposition 1, when Euclid said, “Let ������������ be the given finite straight line,” he assumed that, given any two distinct points there is exactly one line that contains them. Of course, that assumes we have two points! It is best if we assume there are points in the plane as well: Every plane contains at least three noncollinear points. Euclid continued on to show that the measures of each of the three sides of his triangle were equal. It makes sense to discuss the measure of a segment in terms of distance. To every pair of points ������ and ������ there corresponds a real number ������������������������(������, ������) ≥ ������, called the distance from ������ to ������. Since the distance from ������ to ������ is equal to the distance from ������ to ������, we can interchange ������ and ������: ������������������������(������, ������) = ������������������������(������, ������). Also, ������ and ������ coincide if and only if ������������������������(������, ������) = ������. Using distance, we can also assume that every line has a coordinate system, which just means that we can think of any line in the plane as a number line. Here’s how: given a line, ������, pick a point ������ on ������ to be “0” and find the two points ������ and ������ such that ������������������������(������, ������) = ������������������������(������, ������) = ������. Label one of these points to be 1 (say point ������), which means the other point ������ corresponds to -1. Every other point on the line then corresponds to a real number determined by the (positive or negative) distance between 0 and the point. In particular, if after placing a coordinate system on a line, if a point ������ corresponds to the number ������, and a point ������ corresponds to the number ������, then the distance from ������ to ������ is ������������������������(������, ������) = |������ − ������|. History of Geometry: Examine the site http://geomhistory.com/home.html to see how geometry developed over time.

Relevant Vocabulary (3 minutes)
The terms point, line, plane, distance along a line, and distance around a circular arc are all left as undefined terms; that is, they are only given intuitive descriptions. For example, a point can be described as a location in the plane, and a straight line can be said to extend in two directions forever. It should be emphasized that, while we give these terms pictorial representations (like drawing a dot on the board to represent a point), they are concepts, and they only exist in the sense that other geometric ideas depend on them.
Relevant Vocabulary Geometric Construction: A geometric construction is a set of instructions for drawing points, lines, circles, and figures in the plane. The two most basic types of instructions are the following: 1. 2. Given any two points ������ and ������, a ruler can be used to draw the line ������������������ or segment ������������. (Abbreviation: Draw ������������.) Given any two points ������ and ������, use a compass to draw the circle that has center at ������ that passes through ������. (Abbreviation: Draw circle: center ������C, radius ������������.)

Constructions also include steps in which the points where lines or circles intersect are selected and labeled. (Abbreviation: Mark the point of intersection of the lines ������������ and ������������ by ������, etc.) Figure: A (two-dimensional) figure is a set of points in a plane. Usually the term figure refers to certain common shapes like triangle, square, rectangle, etc. But the definition is broad enough to include any set of points, so a triangle with a line segment sticking out of it is also a figure. Equilateral Triangle: An equilateral triangle is a triangle with all sides of equal length.

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NYS COMMON CORE MATHEMATICS CURRICULUM

Lesson 1

M1

GEOMETRY

Collinear: Three or more points are collinear if there is a line containing all of the points; otherwise, the points are non collinear. Length of a Segment: The length of the segment ������������ is the distance from ������ to ������ and is denoted |������������| or ������������. Thus, ������������ = ������������������������(������, ������).

In this course, you will have to write about distances between points and lengths of segments in many , if not most, homework problems. Instead of writing ������������������������(������, ������) all of the time, which is rather long and clunky notation, we will instead use the much simpler notation ������������ for both distance and length of segments. You may have already noticed that ������������ can stand for a line or a segment. From now on it can also stand for a number (the distance between A and B). Which one do we mean? At first, navigating the different uses of this notation may seem confusing, but the context will help you quickly decide how the notation ������������ is being used. Here are some examples:     The line ������������ intersects… ������������ + ������������ = ������������ Find ������������ so that ������������ ∥ ������������. ������������ = ������. ������������ refers to a line. Only numbers can be added, so ������������ is a length or distance. Only figures can be parallel, so ������������ is a line or segment. ������������ refers to the length of the segment AB or the distance from ������ to ������.

When the context is not clear or formality is important, you should use the standard notations for segments, lines, rays, distances, and lengths:      A ray with vertex ������ that contains the point ������: A line that contains points ������ and ������: A segment with endpoints ������ and ������: The length of segment ������������: The distance from ������ to ������: ������������. ⃡ ������������ or ������������������ . ������������. |������������|. ������������������������(������, ������).

Coordinate System on a Line: Given a line ������, a coordinate system on ������ is a correspondence between the points on the line and the real numbers such that (i) to every point on ������ there corresponds exactly one real number, (ii) to every real number there corresponds exactly one point of ������, and (iii) the distance between two distinct points on ������ is equal to the absolute value of the difference of the corresponding numbers.

Exit Ticket (3 minutes)

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NYS COMMON CORE MATHEMATICS CURRICULUM

Lesson 1

M1

GEOMETRY

Name ___________________________________________________

Date____________________

Lesson 1: Construct an Equilateral Triangle
Exit Ticket
We saw two different scenarios where we used the construction of an equilateral triangle to help determine a needed location (i.e., the friends playing catch in the park and the sitting 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.

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NYS COMMON CORE MATHEMATICS CURRICULUM

Lesson 1

M1

GEOMETRY

Exit Ticket Sample Solution
We saw two different scenarios where we used the construction of an equilateral triangle to help determine a needed location (i.e., the friends playing catch in the park and the sitting 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. Students might describe a need to determine the locations of fire hydrants, friends meeting at a restaurant, or parking lots for a stadium, etc.

Problem Set Sample Solutions
1. Write a clear set of steps for the construction of an equilateral triangle. Use Euclid’s Proposition 1 as a guide. 1. 2. 3. 4. Draw circle CJ : center J, radius JS. Draw circle CS: center S, radius SJ. Label the intersection as M. Join S, J, M.

MP.5

2.

Suppose two circles are constructed using the following instructions: Draw circle: Center ������, radius ������������. Draw circle: Center ������, radius ������������. Under what conditions (in terms of distances ������������, ������������, ������������) do the circles have a. One point in common? If AB+CD=AC or AC+AB=CD or AC+CD=AB. Ex.

b.

No points in common? If AB+CD < AC or AB +AC < CD or CD+AC < AB. Ex.

c.

Two points in common? If AC