TCSS Unit 2 Information

Georgia Milestones Domain & Weight: 35%

Curriculum Map: Similarity, Congruence, and Proofs

Content Descriptors: Concept 1: Understand similarity in terms of similarity

transformations. Concept 2: Prove theorems involving similarity. Concept 3: Understand congruence in terms of rigid motions. Concept 4: Prove Geometric Theorems. Concept 5: Make geometric constructions.

Content from Frameworks: Similarity, Congruence, and Proofs

Unit Length: Approximately 49 days

Georgia Milestones Study Guide for TCSS Unit 2

2017- 2018

TCSS GSE Geometry Unit 2 Curriculum Map

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Unit Rational: Building on standards from Coordinate Algebra and from middle school, students will use transformations and proportional reasoning to develop a formal understanding of similarity and congruence. Students will identify criteria for similarity and congruence of triangles, develop facility with geometric proofs (variety of formats), and use the concepts of similarity and congruence to prove theorems involving lines, angles, triangles, and other polygons.

Prerequisites: As identified by the GSE Frameworks Understand and use reflections, translations, and rotations. Define the following terms: circle, bisector, perpendicular and parallel. Solve multi-step equations. Understand angle sum and exterior angle of triangles. Know angles created when parallel lines are cut by a transversal. Know facts about supplementary, complementary, vertical, and adjacent angles. Solve problems involving scale drawings of geometric figures. Draw geometric shapes with given conditions. 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. Draw polygons in the coordinate plane given coordinates for the vertices.

Length of Unit 49 Days

Concept 1 Concept 2 Concept 3 Concept 4 Concept 5 Understand

similarity in terms of similarity transformations.

Prove theorems involving similarity.

Understand congruence in terms of

rigid motions. Prove geometric

Theorems. Make

geometric constructions.

GSE Standards GSE Standards GSE Standards GSE Standards GSE Standards

MGSE9-12.G.SRT.1 Verify experimentally the properties of dilations given by a center and a scale factor:

a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.

b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

MGSE9-12.G.SRT.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally (and its converse); the Pythagorean Theorem using triangle similarity.

MGSE9-12.G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

MGSE9-12.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.

MGSE9-12.G.CO.7 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.

MGSE9-12.G.CO.9 Prove 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.

MGSE9-12.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

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MGSE9-12.G.SRT.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

MGSE9-12.G.SRT.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

CC9-12.G.CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. (Extend to include HL and AAS)

MGSE9-12.G.CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 degrees; 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.

MGSE9-12.G.CO.11 Prove 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.

MGSE9-12.G.GPE.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; (Focus on quadrilaterals, right triangles, and circles.)

constructing a line parallel to a given line through a point not on the line.

MGSE9-12.G.CO.13 Construct an equilateral triangle, a square, and a regular hexagon, each inscribed in a circle.

Lesson Essential Question Lesson Essential Question Lesson Essential Question Lesson Essential Question

Lesson Essential Question

• What is a dilation and how does this transformation affect a figure in the coordinate plane?

• What strategies can I use to determine missing side lengths and areas of similar figures?

• Under what conditions are similar figures congruent?

• How do I know which method to use to prove two triangles congruent?

• How do I know which method to use to prove two triangles similar?

• In what ways can I use congruent triangles to justify many geometric constructions?

• How do I make geometric

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• How do I prove geometric theorems involving lines, angles, triangles, and parallelograms?

constructions?

Vocabulary Vocabulary Vocabulary Vocabulary Vocabulary Dilations Center Scale Factor Parallel lines Line Segments Ratio Similarity Transformations Corresponding angles Corresponding sides Proportionality AA criterion

Adjacent Angles Alternate Exterior Angles Alternate Interior Angles Angle Bisector Centroid Circum center Coincidental Complementary Angles Congruent Congruent Figures Corresponding Angles Corresponding Sides Dilation Parallel Pythagorean Theorem Endpoints Equiangular Similarity

Equilateral Exterior Angle of a Polygon In center Intersecting Lines Intersection Line Line Segment or Segment Linear Pair Measure of each Interior Angle of a Regular n-gon: Orthocenter Parallel Lines Perpendicular Lines Plane Point Proportion Ratio Ray Rigid motions Transform Corresponding Angles Corresponding Sides

Reflection Reflection Line Regular Polygon Remote Interior Angles of a Triangle Rotation Same-Side Interior Angles Same-Side Exterior Angles Scale Factor Similar Figures Skew Lines Sum of the Measures of the Interior Angles of a Convex Polygon Supplementary Angles Transformation Translation Transversal Vertical Angles Alternate interior Perpendicular bisector Equidistant Endpoints Theorems: Interior angle sum Theorem Base angles of Isosceles Triangle Theorem Segments of midpoints of a triangle Theorem Medians of a triangle Theorem Median Isosceles Triangle Midpoints

Construction Segments Angles Bisect Perpendicular lines Perpendicular bisectors Parallel lines Equilateral triangle Regular hexagon inscribed

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Sample Assessment Items

Concept 1 Sample Assessment Items

Concept 2 Sample Assessment Items

Concept 3 Sample Assessment Items

Concepts 4 Sample Assessment Items

Concepts 5 MGSE9-12.G.SRT.1 MGSE9-12.G.SRT.4

Justify the last two steps of the proof.

Given: &

Prove: R

S T

U

a. Symmetric Property of ; SSS b. Reflexive Property of ; SAS

c. Reflexive Property of ; SSS

d. Symmetric Property of ; SAS MGSE9-12.G.SRT.5 In the triangle below, what is the approximate value of x?

MGSE9-12.G.CO.6 MGSE9-12.G.CO.9

What can be concluded if ∠1 ≅ ∠7?

a. t ⊥ p

b. p ⊥ q

c. p Π q d. p Π t

MGSE9-12.G.CO.10 Match each statement in the proof with the correct reason. A. Definition of midpoint B. SAS congruence criterion C. Corresponding parts of

congruent triangles are congruent.

D. SSS congruence criterion E. Definition of

perpendicular bisector F. Given G. If a point is on the

bisector of an angle, it is equidistant from the sides of the angle.

MGSE9-12.G.CO.12 In the coordinate plane Parallelogram FGHJ was Which diagram below shows segment M’N’ is the result of a dilation of segment MN by a scale factor of ⅓. Which point is the center of dilation?

translated 3 units down to form parallelogram F’G’H’J’. Parallelogram F’G’H’J’ was then rotated 90° counterclockwise about point G’ to obtain parallelogram

a correct mathematical construction using only a compass and a straightedge to bisect an angle?

A. (1, 3) F”G”H”JJ”. B. (0, 0) C. (– 5 , 0)

a.

D. (– 4 , 1)

b.

MGSE9-12.G.SRT.2

Which transformation results in a figure that is similar to the original figure but has a greater perimeter?

Which statement is true about parallelogram FGHJ and parallelogram F’G’H’J’?

c.

a. a dilation of by a a. The figures are similar but

scale factor of 0.25 not congruent. b. a dilation of by a scale factor of 0.5 c. a dilation of by a

b. The figures are congruent but not similar.

d.

scale factor of 1 c. The figures are both similar d. a dilation of by a and congruent. scale factor of 2

d. The figures are neither MGSE9-12.G.SRT.3 similar nor congruent. Which method would it be possible to prove the triangles below are similar (if

1. 1. Given

2. 2. Given

3. 3. 4. 4.

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TCSS GSE Geometry Unit 2

BC

possible)? MGSE9-12.G.CO.7

For which pair of triangles would you use AAS to prove the congruence of the 2 triangles?

H. Reflexive property of congruence

MGSE9-12.G.CO.13 What is the first step when inscribing a regular hexagon in the circle below?

a. SSS b. AA c. SAS d Not similar

a. 4 in b. 4.5 in c. 4.9 in d. 5.1 in

A)

B)

C)

D)

MGSE9-12.G.CO.8 Which statement is true about ABC and FED?

a. The triangles are similar, but not congruent. b. The triangles are congruent using ASA. c. The triangles are congruent

using SAS.

Given: AB ≅ AC, M is the midp

Prove: ∠B ≅ ∠C

1. AB ≅ AC

2. BM ≅ MC

3. AM ≅ AM 4. ABM ≅ ACM 5. ∠B ≅ ∠C

Answers: 1.F, 2. A, 3. H, 4.D,

MGSE9-12.G.CO.11 In the accompanying diagram of rhombus ABCD, m∠CAB = 35° What is m∠CDA?

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d. The triangles are congruent using SSS.

MGSE9-12.G.GPE.4 Given A (2,3), B (5, -1), C (1, 0), D (-4, -1), E (0,2), F (-1,-2)

Prove: ∠ABC≅∠DEF Plot the points on a coordinate plane.

Use the Distance Formula to find the lengths of the sides of each triangle.

Resources – Concept 1 Resources – Concept 2 Resources – Concept 3 Resources – Concept 4 Resources – Concept 5 Instructional

Strategies and Common Misconceptions

Dilations practice (G.SRT.1)

Dilation drills G.SRT.1) Similar polygons

notes (G.SRT.2) Similar triangles

Graphic Organizer (G.SRT.2) Key

Instructional Strategies and Common Misconceptions

Congruence Graphic Organizer (G.SRT.5)

Property/Postulate/ T heorem “Cheat Sheet”

Properties matching card game (activator)

Triangle Bisector notes (G.SRT.5)

Instructional Strategies and Common Misconceptions

Congruency notes (with proofs) (G.CO.7)

Practice problems with application (G.CO.7)

Congruency Lesson Packet (G.CO.7)

Instructional Strategies and Common Misconceptions

Review of Parallel Lines & Transversals

Properties of Parallelograms notes (G.CO.11)

Proving properties of parallelograms activity (G.CO.11)

Floor patterns extension

ti it

Instructional Strategies and Common Misconceptions

Constructing lines/bisecting angles notes and practice (G.CO.12)

How to bisect an angle (G.CO.12)

How to copy an angle (G.CO.12)

How to construct a line parallel to

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TCSS GSE Geometry Unit 2

These tasks were taken from the

GSE Frameworks. Similar Triangles

Activator/Summarize r (G.SRT.1,2,3) Teacher Student

Proving Similar Triangles – guided notes (G.SRT.3) Teacher Student

Textbook Resources Holt McDougal –

Explorations in Core Math p 181- 184(G.SRT.1)

Holt McDougal – Explorations in Core Math p 155-165 &168-170(G.SRT.2)

Holt McDougal – Explorations in Core Math p 171- 176(G.SRT.3)

These tasks were taken from the

GSE Frameworks. Shadow Math –

extension activity (G.SRT.2,3,5)

Pythagorean Theorem using Triangle Similarity (G.SRT.4)

Textbook Resources Holt McDougal –

Explorations in Core Math p 97-108, 177- 180 (G.SRT.4)

Holt McDougal – Explorations in Core Math p 90,96,102(G.SRT.5)

These tasks were taken from the

GSE Frameworks. Introducing

Congruence activator/summarizer (G.CO.6&7) Teacher Student

Proving Triangles are Congruent station activity (G.CO.6-8) Teacher Student

Textbook Resources Holt McDougal –

Explorations in Core Math p65-70 (G.CO.6)

Holt McDougal – Explorations in Core Math p77-82 (G.CO.7)

Holt McDougal – Explorations in Core Math p83,86- 96 (G.CO.8)

These tasks were taken from the

GSE Frameworks. Lunch lines

notes/practice (G.CO.9) Teacher Student

Centers of Triangles (G.CO.10) Teacher Student

Proving Quadrilaterals in the Coordinate Plane (G.CO.11) Teacher Student

Culminating Task: Geometry Gardens – project (extension activity)

Textbook Resources Holt McDougal –

Explorations in Core Math p35-54, 60-61, 84-85, 88 (G.CO.9) *not numbers 1-3 on page 37 or 1- 2 on page 38

Holt McDougal – Explorations in Core Math p122-130 (G.CO.10)

Holt McDougal – Explorations in Core Math p131-154 (G.CO.11)

another line (G.CO.12) How to construct a

perpendicular line (G.CO.12)

Constructions worksheet (G.CO.12)

Constructions practice (G.CO.12)

How to construct a hexagon and an equilateral triangle inscribed in a circle (G.CO.13)

Textbook Resources Holt McDougal –

Explorations in Core Math p5-16, 55-59, 62-64 (G.CO.12) Holt McDougal – Explorations in Core Math p115- 120 (G.CO.13)

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Differentiated Activities

Concept 1 Differentiated Activities

Concept 2 Differentiated Activities

Concept 3 Differentiated Activities

Concept 4 Differentiated Activities

Concept 5 Dilations in the

Coordinate Plane (G.SRT.1&2) Teacher Student

Vocabulary/Postulate Card match activity

Proof Scramble

Formalizing Triangle Congruence (G.CO.8) Teacher Student

Floodlights FAL - application (G.SRT.4&5, G.CO.9-11) – extension activity

Resources recommended for Math Support

Resources recommended for Math Support

Resources recommended for Math Support

Resources recommended for Math Support

Resources recommended for Math Support

Interactive Vocabulary Site (differentiate how vocabulary is presented)

What is similarity (activator)

G.SRT.1 GADOE teacher notes

G.SRT.2 GADOE teacher notes

G.SRT.3 GADOE teacher notes

These tasks were taken from the

GSE Frameworks.

Intro. activity

Triangle Sum notes

These tasks were taken from the

GSE Frameworks.

Triangle Proportionality Theorem (G.SRT.2-5) Teacher Student

Analyzing Congruence Proofs FAL

G.CO.6 GADOE teacher notes

G.CO.7 GADOE teacher notes

These tasks were taken from the

GSE Frameworks.

Evaluating Statements about length & area FAL(G.CO.9-11)

G.CO.12 GADOE teacher notes

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At the end of Unit 2 stud ent’s should be able to say “I can…” Take Away

Given a center and a scale factor, verify experimentally, that when dilating a figure in a coordinate plane, a segment of the pre-image that does not pass through the center of the dilation, is parallel to its image when the dilation is performed. However, a segment that passes through the center remains unchanged.

Given a center and a scale factor, verify experimentally, that when performing dilations of a line segment, the pre- image, the segment which becomes the image is longer or shorter based on the ratio given by the scale factor.

Use the idea of dilation transformations to develop the definition of similarity. Given two figures determine whether they are similar and explain their similarity based on the equality of

corresponding angles and the proportionality of corresponding sides. Use the properties of similarity transformations to develop the criteria for proving similar triangles: AA. Use AA, SAS, SSS similarity theorems to prove triangles are similar. Prove a line parallel to one side of a triangle divides the other two proportionally, and its converse. Prove the Pythagorean Theorem using triangle similarity. Use similarity theorems to prove that two triangles are congruent. Use descriptions of rigid motion and transformed geometric figures to predict the effects rigid motion has on

figures in the coordinate plane. Knowing that rigid transformations preserve size and shape or distance and angle, use this fact to connect the idea

of congruency and develop the definition of congruent. Use the definition of congruence, based on rigid motion, to show two triangles are congruent if and only if their

corresponding sides and corresponding angles are congruent. Use the definition of congruence, based on rigid motion, to develop and explain the triangle congruence criteria:

ASA, SSS, and SAS. Prove vertical angles are congruent. Prove when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are

congruent. Prove points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s

endpoints. Prove the measures of interior angles of a triangle have a sum of 180º. Prove base angles of isosceles triangles are congruent. Prove the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length. Prove the medians of a triangle meet at a point.

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Prove properties of parallelograms including: opposite sides are congruent, opposite angles are congruent, diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

Copy a segment and an angle. Bisect a segment and an angle. Construct perpendicular lines, including the perpendicular bisector of a line segment. Construct a line parallel to a given line through a point not on the line. Construct an equilateral triangle so that each vertex of the equilateral triangle is on the circle. Construct a square so that each vertex of the square is on the circle. Construct a regular hexagon so that each vertex of the regular hexagon is on the circle.