# Congruent Segments Definition (I)

[color=#000000]The following applet illustrates what it means for segments to be classified as [b]congruent segments.  [br][br][/b]Interact with the applet below for a few minutes.  Be sure to move the locations of the black points around each time [i]before[/i] you drag the slider!  [br][br]Then answer the questions that follow.  [/color]
[color=#000000][b]Questions:  [/b][br][br]1) What transformations have we learned about thus far?  List them.[br][br]2) What transformations did you observe here while interacting with this applet above?  [br][br]3) How would you describe what it means for segments to be classified as [b]congruent [br]    segments [/b]with respect to any one (or more) of the transformations you observed here [br]    in this applet?  Explain.  [/color]

# Translations and Rotations

##### Direct Isometries
Translations and rotations are [i]direct isometries[/i]. If you read the names of the vertices in cyclic order (A-B-C and A'-B'-C'), both would be read in the counterclockwise (or clockwise) direction.

# Are the triangles congruent (part 2)?

 Use the given measurement tools to establish that the corresponding sides of triangle ABS and triangle A'B'C' are parallel and have equal lengths. Use the given transformation tools to establish that triangle ABC is congruent to triangle A'B'C'.

# Congruent Figures: Dynamic Illustration

[color=#0000ff]Recall an ISOMETRY is a transformation that preserves distance.[/color] So far, we have already explored the following isometries:[br][br][color=#0000ff]Translation by Vector[br]Rotation about a Point[br]Reflection about a Line[br]Reflection about a Point ( same as 180-degree rotation about a point) [/color][br][br]For a quick refresher about [color=#0000ff]isometries[/color], see this [url=https://www.geogebra.org/m/KFtdRvyv]Messing with Mona applet[/url].
##### CONGRUENT FIGURES
[b]Definition: [br][br]Any two figures are said to be CONGRUENT if and only if one can be mapped perfectly onto the other using [color=#0000ff]any 1 or composition of 2 (or more) ISOMETRIES.[/color][/b][br][br]The applet below dynamically illustrates, [b]by DEFINITION[/b], what it means for any 2 figures (in this case, triangles) to be [b]CONGRUENT.[/b] [br][br]Feel free to move the BIG WHITE VERTICES of either triangle anywhere you'd like at any time.

# SAS: Dynamic Proof!

[color=#000000]The [/color][b][u][color=#0000ff]SAS Triangle Congruence Theorem[/color][/u][/b][color=#000000] states that [/color][b][color=#000000]if 2 sides [/color][color=#000000]and their [/color][color=#ff00ff]included angle [/color][color=#000000]of one triangle are congruent to 2 sides and their [/color][color=#ff00ff]included angle [/color][color=#000000]of another triangle, then those triangles are congruent. [/color][/b][color=#000000]The applet below uses transformational geometry to dynamically prove this very theorem. [br][br][/color][color=#000000]Interact with this applet below for a few minutes, then answer the questions that follow. [br][/color][color=#000000]As you do, feel free to move the [b]BIG WHITE POINTS[/b] anywhere you'd like on the screen! [/color]
[color=#000000][b]Questions:[/b][br][br]1) What geometry transformations did you observe in the applet above? List them. [br]2) What common trait do all these transformations (you listed in your response to (1)) have? [br]3) Go to [url=https://www.geogebra.org/m/d9HrmyAp#chapter/74321]this link[/url] and complete the first 5 exercises in this GeoGebra Book chapter. [/color]

# CCSS HS GEOMETRY RESOURCES!!!

[color=#0000ff]This worksheet contains [/color][color=#980000][b]links[/b][/color][color=#0000ff] to [/color][b]HUNDREDS [/b][b]of dynamic and engaging geometry resources. [/b][b][color=#0000ff]Each worksheet is mapped to 1 (or more) of the standards listed in the [/color][url=http://www.corestandards.org/Math/Content/HSG/introduction/]HIGH SCHOOL: Geometry[/url] [color=#0000ff]section of the [/color][url=http://www.corestandards.org/Math/]Common Core State Standards Initiative for Mathematics[/url][color=#980000]. [/color][color=#274e13] [br][/color][/b][br][color=#980000][b]LINKS: [br][/b][/color][br][url=https://www.geogebra.org/m/z8nvD94T#chapter/0]CCSS High School: Geometry (Congruence) Volume 1[/url][br][br][url=https://www.geogebra.org/m/munhXmzx#chapter/0]CSSS High School: Geometry (Congruence) Volume 2[/url][br][br][url=https://www.geogebra.org/m/dPqv8ACE#chapter/0]CCSS High School: Geometry (Similarity, Right Triangles, & Trigonometry)[/url] [br][br][url=https://www.geogebra.org/m/C7dutQHh#chapter/0]CCSS High School: Geometry (Circles)[/url][br][br][url=https://www.geogebra.org/m/K2YbdFk8#chapter/0]CCSS High School: Geometry (Expressing Geometric Properties with Equations)[/url][br][url=https://www.geogebra.org/m/xDNjSjEK#chapter/0][br]CCSS High School: Geometry (Geometric Measurement & Dimension)[/url][br][br][url=https://www.geogebra.org/m/pptbYhsy#chapter/0]CCSS High School: Geometry (Modeling with Geometry)[/url]﻿[br][br][b][color=#0000ff]*NEW![/color][/b] [url=https://www.geogebra.org/m/NjmEPs3t]CCSS High School: Geometry (Higher Level Enrichment Challenges)[/url]
##### SAMPLE 2: What 2 theorems are dynamically being illustrated below? (Feel free to move the white points wherever you'd like.)
[color=#000000]Teachers can use these resources as a powerful means to naturally [br][br][/color][b][color=#0000ff]1) Foster Discovery Learning[br][/color][color=#0000ff]2) Provide Meaningful Remediation[br]3) Differentiate Instruction, &[br]4) Assess students' understanding.[/color][color=#000000] [/color][/b][br][br][color=#000000]Since any curriculum is [/color][b][i][color=#980000]always[/color][/i][/b][color=#000000] a fluid document, these books, too, will continue to remain works in progress.[/color][br][br][b][color=#0000ff]Teachers:[/color][/b][color=#000000] [br]It is my hope that these resources help empower your students to actively (and regularly) discover the fascinating world of mathematics around them. [br][/color][br][b][color=#0000ff]Students:[/color][/b][color=#000000] [br]It is my hope that these resources help you discover & help reinforce mathematics concepts in a way that makes sense to you. [/color]
[b][color=#980000]These GeoGebra books display the amazing work from several esteemed members of the GeoGebra community. I am truly humbled and amazed by their talents. These comprehensive resources would not have been possible without their contributions. [br][br]I would like to express a [u]HUGE THANK YOU[/u] to[br][br][/color][/b][url=https://www.geogebra.org/orchiming]Anthony C.M. OR[/url][br][url=https://www.geogebra.org/stevephelps]Steve Phelps[/url][br][url=https://www.geogebra.org/jennifer+silverman]Jennifer Silverman[/url][br][url=https://www.geogebra.org/tedcoe]Dr. Ted Coe[/url][br][url=https://www.geogebra.org/scruz10]Samantha Cruz[/url][br][url=https://www.geogebra.org/tlindy]Terry Lee Lindenmuth[br][/url][url=https://www.geogebra.org/ra%C3%BAl+falc%C3%B3n]Raul Manuel Falcon Ganfornina[/url] [br][url=https://www.geogebra.org/walch+education]Walch Education[/url]﻿[br][url=https://www.geogebra.org/edc+in+maine#]EDC in Maine[/url][br][br]For questions, suggestions, and/or comments, feel free to e-mail me at any time. [br]I wish you much success in your journey of teaching and/or learning mathematics! [br][br]Best,[br][br][url=https://www.geogebra.org/tbrzezinski]Tim Brzezinski[br][br][/url][color=#1e84cc]Independent Mathematics Education Consultant ([url=http://www.dynamicmathsolutions.com/]Dynamic Math Solutions[/url])[br][/color][color=#1e84cc]Adjunct Mathematics Instructor at Central Connecticut State University[br]Former High School Mathematics Teacher (15 years) at Berlin High School (CT, USA)[/color][br][br]E-Mail: dynamicmathsolutions@gmail.com [br]Twitter: [url=https://twitter.com/dynamic_math]@dynamic_math[/url][br]

# CCSS HS FUNCTIONS RESOURCES!!!

﻿This worksheet currently contains [color=#0000ff][b]links[/b][/color] to [b]over 150 [/b][b]DYNAMIC and ENGAGING resources pertaining to FUNCTIONS. [/b][b]Each worksheet is mapped to 1 (or more) of the standards listed in the[color=#0000ff] [url=http://www.corestandards.org/Math/Content/HSF/introduction/]CCSS High School: Functions[/url]﻿ [/color][/b]section of the[b] [color=#cc0000][url=http://www.corestandards.org/Math/]Common Core State Standards Initiative for Mathematics[/url].[/color][/b][color=#666666][b] [/b][br][br][/color][color=#980000][b]LINKS: [br][/b][/color][br][url=https://www.geogebra.org/m/k6Dvu9f3]CCSS High School: Functions (Interpreting Functions)[/url]﻿[br][br][url=https://www.geogebra.org/m/uTddJKRC]CCSS High School: Functions (Building Functions)[/url][br][br][url=https://www.geogebra.org/m/GMvvpwrm]CCSS High School: Functions (Linear, Quadratic, & Exponential Models)[/url][br][br][url=https://www.geogebra.org/m/aWuJMDas]CCSS High School: Functions (Trigonometric Functions)[/url][br][br][br][color=#000000]Teachers can use these resources as a powerful means to naturally [br][br][/color][b][color=#0000ff]1) Foster Discovery Learning[br][/color][color=#0000ff]2) Provide Meaningful Remediation[br]3) Differentiate Instruction, &[br]4) Assess students' understanding.[/color][color=#000000] [br][br][/color][/b][color=#000000]Since any curriculum is [/color][b][i][color=#0000ff]always[/color][/i][/b][color=#000000] a fluid document, these books, too, will continue to remain works in progress. More items will continue to be added to these volumes. [br][/color]
##### Sample 2 - ODD FUNCTION ILLUSTRATOR: Feel free to move any of the BIG points wherever you'd like.
[b][color=#0000ff]Teachers:[/color][/b][color=#000000] [br]It is my hope that these resources help empower your students to actively (and regularly) discover the fascinating world of mathematics around them. [br][br][/color][b][color=#0000ff]Students:[/color][/b][color=#000000] [br]It is my hope that these resources help you discover & help reinforce mathematics concepts in a way that makes sense to you.[br][br][/color][b]I would like to express a HUGE THANK YOU to[/b] [url=https://www.geogebra.org/orchiming][color=#0000ff]Anthony C.M.Or[/color][/url] [b]and[/b] [color=#0000ff][url=https://www.geogebra.org/stevephelps]Steve Phelps[/url], [/color][br][b]whose work also appears in this project. [/b][br][br]For questions, suggestions, and/or comments, feel free to e-mail me at any time. [br]I wish you much success in your journey of teaching and/or learning mathematics! [br][br]Best,[br][br][url=https://www.geogebra.org/tbrzezinski]Tim Brzezinski[br][/url][color=#1e84cc][br]Independent Mathematics Education Consultant ([url=http://www.dynamicmathsolutions.com/]Dynamic Math Solutions[/url])[br][/color][color=#1e84cc]Adjunct Mathematics Instructor at Central Connecticut State University[br]Former High School Mathematics Teacher (15 years) at Berlin High School (CT, USA)[/color][br][br]E-Mail: dynamicmathsolutions@gmail.com [br]Twitter: [url=https://twitter.com/dynamic_math]@dynamic_math[/url]