APS Geometry Unit 1 Book

Wendy Sanchez, badboy_wit03, May 9, 2019

MGSE9-12.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. MGSE9-12.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). MGSE9-12.G.CO.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. MGSE9-12.G.CO.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. MGSE9-12.G.CO.5 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. Make geometric constructions. 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.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.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. MGSE9-12.G.CO.13 Construct an equilateral triangle, a square, and a regular hexagon, each inscribed in a circle. Understand congruence in terms of rigid motions

Table of Contents

  1. MSGE9-12.G.CO.1
    1. Collinear Points (Definition Prompt) APS
    2. Parallel Lines Definition APS
    3. Point / Midpoint / Segment Definition APS
  2. MGSE9-12.G.CO.2
    1. Exploring Rotations (Intervals of 90 degrees only) APS
    2. Transformation Station Rotation APS
    3. Transformation Station Reflection APS
    4. Transformation Station Translation APS
  3. MSGE9-12.G.CO.3
    1. Exploration: Rotating A Figure Onto Itself APS
    2. Symmetries of Regular Polygons - Rotational Symmetry APS
    3. Reflect onto Itself 1 APS
    4. Reflect onto Itself 2 APS
    5. Reflect onto Itself 3 APS
  4. MGSE9-12.G.CO.4
    1. Transformations and Parallel Lines APS
  5. MGSE9-12.G.CO.5
    1. Reflections in the Coordinate Plane APS
    2. Rotations in the Coordinate Plane APS
    3. Translating by Vector <a,b> APS
  6. MGSE9-12.G.CO.7
    1. Congruence Transformations APS
  7. MGSE9-12.G.CO.8
    1. SAS - Exercise 1A APS
  8. MGSE9-12.G.CO.12
    1. Copying a Segment APS
    2. Compass - Straightedge Construction - Copying a Segment APS
    3. Construction: Copy an Angle APS
    4. Bisecting an Angle APS
    5. Construction: Bisect an angle APS
    6. Bisecting a Segment 2 APS
    7. Construction: Perpendicular Bisector of a Line Segment - APS
    8. Segment Bisector (Midpoint) APS
  9. MGSE9-12.G.CO.13
    1. Equilateral Triangle inscribed in a circle construction APS
    2. Equilateral Triangle Construction (Dynamic Illustration) APS
    3. Equilateral Triangle Construction Template APS
    4. Constructions: Square Inscribed in a Circle APS
    5. Square Construction APS
    6. Construction: Regular Hexagon Inscribed in a Circle APS

1.

MSGE9-12.G.CO.1

  • 1. Collinear Points (Definition Prompt) APS
  • 2. Parallel Lines Definition APS
  • 3. Point / Midpoint / Segment Definition APS

1.1.

Collinear Points (Definition Prompt) APS

Warm Up--See below:
Questions:[br][br]1) After seeing what you now see, what does it mean for points to be non-collinear? [br]2) Consider [b]only 2 points[/b] [b]A[/b] and [b]B[/b]. Is it ever possible for [b]A[/b] & [b]B[/b] to be non-collinear? Why or why not? Explain.
– badboy_wit03, Joseph Marutollo, Tim Brzezinski

1.2.

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1.3.

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2.

MGSE9-12.G.CO.2

  • 1. Exploring Rotations (Intervals of 90 degrees only) APS
  • 2. Transformation Station Rotation APS
  • 3. Transformation Station Reflection APS
  • 4. Transformation Station Translation APS

2.1.

Exploring Rotations (Intervals of 90 degrees only) APS

Rotations
When Rotating points 90 degrees CCW, describe how the corresponding point move with regard to the center of rotation.
When Rotating points 180 degrees CCW, describe how the corresponding point move with regard to the center of rotation.
– badboy_wit03, Wendy Sanchez, Rick Stuart

2.2.

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2.3.

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2.4.

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3.

MSGE9-12.G.CO.3

  • 1. Exploration: Rotating A Figure Onto Itself APS
  • 2. Symmetries of Regular Polygons - Rotational Symmetry APS
  • 3. Reflect onto Itself 1 APS
  • 4. Reflect onto Itself 2 APS
  • 5. Reflect onto Itself 3 APS

3.1.

Exploration: Rotating A Figure Onto Itself APS

How many times did the triangle map onto itself during the course of the 360 degree rotation?
What is the minimum number of degrees for the triangle to be mapped onto itself?
How many times did the square map onto itself during the course of the 360 degree rotation?
What is the minimum number of degrees for the square to be mapped onto itself?
How many times did the pentagon map onto itself during the 360 degree rotation?
What is the minimum number of degrees in order for the pentagon to map onto itself?
Using all of the above data, can you produce a formula that would find the minimum number of degrees needed in order for any regular polygon to be mapped onto itself?
– S Sharma, Kevin Farrell, Dan Hayward

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3.3.

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3.4.

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3.5.

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4.

MGSE9-12.G.CO.4

  • 1. Transformations and Parallel Lines APS

4.1.

Transformations and Parallel Lines APS

This applet lets you explore congruent angles formed by parallel lines with transformations.
1) Are the lines parallel? How do you know?[br]2) Drag points B and C to show that lines remain parallel.[br]3) Click “Step 1.” What transformation maps <HBD onto <H’BD’? [br] a) 180o Rotation b) Reflection c) Translation[br]4) Click “Step 2.” What transformation maps <D’BH’ onto <D”CH”?[br]a) 180o Rotation b) Reflection c) Translation[br]5) Click “Step 3.” What transformation maps <D”CH” onto <D’’’CH’’’?[br]6) a) 180o Rotation b) Reflection c) Translation[br]7) Drag points B and C to show that angles remain congruent. Drag points D and H to show that the transformations maintain congruence.[br]EXAMPLE: Write a paragraph proof. Prove that <1 is congruent to <3 using transformations.[br]A 180o Rotation maps <1 onto <2 therefore <1 <2. A translation maps <2 onto <3 therefore <2 <3. By transitive property <1 s congruent to <3.[br]8) Write a paragraph proof, using transformations, proving <2 s congruent to <4.
– Mrs. Tatum, Rachel Fruin

5.

MGSE9-12.G.CO.5

  • 1. Reflections in the Coordinate Plane APS
  • 2. Rotations in the Coordinate Plane APS
  • 3. Translating by Vector <a,b> APS

5.1.

Reflections in the Coordinate Plane APS

1) Plot the points ([b]type[/b] the ordered pair into the [b]input bar[/b], bottom of the page)[br][b]2) A(-3,2), B (-1,4) and C (-5, 3)[br][br][/b]3) [icon]https://www.geogebra.org/images/ggb/toolbar/mode_polygon.png[/icon]Use the polygon tool to construct the triangle ABC[br][br][b]4) [icon]/images/ggb/toolbar/mode_mirroratline.png[/icon][/b]Use the reflection tool (9th tool over, under the reflection diagram)[br]5) Select the [b]Triangle ABC, then y-axis[/b][br][br][br]
[color=#0000ff]When you're done (or if you're unsure of something), feel free to check by watching the quick silent screencast below the applet. [/color]
Video (silent video)
What are the coordinates of A'B'C'? [br]
2.
1) Plot the points ([b]type[/b] the ordered pair into the [b]input bar[/b], bottom of the page)[br][b]2) A(5,-1), B (3,-1) and C (4, -3)[br][/b][br]3)[icon]https://www.geogebra.org/images/ggb/toolbar/mode_polygon.png[/icon]Use the polygon tool to construct the triangle ABC[br][b]4) [icon]https://www.geogebra.org/images/ggb/toolbar/mode_mirroratline.png[/icon][/b]Use the reflection tool (9th tool over, under the reflection diagram)[br]5) Select the [b]Triangle ABC, then x-axis[/b]
List the points of A"B" and C"
If a Triangle is in quadrant 2, and we reflect it over the [b]x_axis [/b]which quadrant would the image be in?
– Wendy Sanchez, Mrs. Tatum, melissahertel

5.2.

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6.

MGSE9-12.G.CO.7

  • 1. Congruence Transformations APS

6.1.

Congruence Transformations APS

Congruence Transformation 1
Show that A is congruent to B by transforming A to B in a combination of transformations.
Congruence Transformation 2
Show that A is congruent to B by transforming A to B in a combination of transformations.
Congruence Transformation 3
Show that A is congruent to B by transforming A to B in a combination of transformations.
– Wendy Sanchez, Chris Hayden

7.

MGSE9-12.G.CO.8

  • 1. SAS - Exercise 1A APS

7.1.

SAS - Exercise 1A APS

In the applet below, use the tools of transformational geometry to informally demonstrate the SAS Triangle Theorem to be true. [br][br]That is, use the tools of transformational geometry to map the [color=#bf9000][b]yellow triangle[/b][/color] onto the empty triangle. [br][br]Before starting, feel free to adjust any aspect of the [color=#bf9000][b]starting triangle[/b][/color] ([color=#666666][b]tilt[/b][/color], [color=#1e84cc][b]size of the included angle[/b][/color], [b]and the positions of points [/b][i]A[/i][b], [/b][i]B[/i][b], and [/b][i]C[/i]). You can also use the [b]black slider[/b] to [b]change the position of the image (empty) triangle.[/b] [br][br][i]Once you do start, it is recommended that you don't readjust these parameters. [/i]
Question:
Describe how you know the two triangle are congruent.
– Wendy Sanchez, Tim Brzezinski

8.

MGSE9-12.G.CO.12

  • 1. Copying a Segment APS
  • 2. Compass - Straightedge Construction - Copying a Segment APS
  • 3. Construction: Copy an Angle APS
  • 4. Bisecting an Angle APS
  • 5. Construction: Bisect an angle APS
  • 6. Bisecting a Segment 2 APS
  • 7. Construction: Perpendicular Bisector of a Line Segment - APS
  • 8. Segment Bisector (Midpoint) APS

8.1.

Copying a Segment APS

Dahlia Zermeno Copy a segment
Copying a Segment
Follow the steps below to copy segment [i]AB[/i]:[br][br]1. Use the line tool to construct line [i]m[/i] below segment [i]AB[/i][br][br]2. Use the point tool to construct point [i]E[/i] on line [i]m[/i]. Right click on the point to show its label.[br][br]3. Use the Compass tool to measure the length of segment [i]AB[/i], then click on Point [i]E[/i] to center the Compass tool at that point. [br][br]4. Construct Point [i]F[/i] at an intersection of line [i]m[/i] and the circle you just constructed.[br][br]5. Hide everything in your sketch (right-click, Show Object) [b]except[/b] for Points [i]A[/i], [i]B[/i], [i]E[/i] and [i]F[/i] and segment [i]AB[/i]. [br][br]6. Use the Segment tool to construct segment [i]EF[/i]. [br][br]7. Drag things around in your sketch and observe what happens.
– Mrs. Tatum, Dahlia Zermeno, Bill Doherty

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8.3.

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8.4.

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8.5.

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8.6.

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8.7.

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8.8.

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9.

MGSE9-12.G.CO.13

  • 1. Equilateral Triangle inscribed in a circle construction APS
  • 2. Equilateral Triangle Construction (Dynamic Illustration) APS
  • 3. Equilateral Triangle Construction Template APS
  • 4. Constructions: Square Inscribed in a Circle APS
  • 5. Square Construction APS
  • 6. Construction: Regular Hexagon Inscribed in a Circle APS

9.1.

Equilateral Triangle inscribed in a circle construction APS

Equilateral Triangle inscribed in a circle construction
– Wendy Sanchez, Brice Dobler

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9.3.

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9.4.

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9.5.

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9.6.

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