Geometry: Introductory Definitions, Postulates, Theorems

Tim Brzezinski, Sep 13, 2015

Table of Contents

  1. Interactive Definition Prompts
    1. Collinear Points (Definition Prompt)
    2. Parallel Lines Definition
    3. Congruent Segments: Quick Exploration
    4. Midpoint Definition
    5. Congruent Angles: Definition
    6. Angle Bisector Definition (I)
    7. Perpendicular Lines Definition (& Consequence)
    8. Segment Bisectors (Definitions)
    9. Perpendicular Bisector Definition
    10. Complementary Meaning?
    11. Supplementary Angles (Quick Exploration)
    12. Triangle Altitude Illustrator and Definition Writing Prompt
    13. Triangle Median Illustrator & Definition Writing Prompt
  2. Practice Exercises with Definitions
    1. Proving Segments Congruent (Exercise)
    2. Proving Angles Congruent (1)
    3. Proving Angles Congruent (2)
    4. Proving Angles Congruent
  3. Segment Addition Postulate Discovery & Exercises (with Midpoint Questions)
    1. SAP Problem 1 (Discovery)
    2. SAP - Problem 2
    3. SAP - Problem 3
    4. Midpoint Definition Question (I)
    5. Midpoint Definition Question (II)
  4. Angle Addition Postulate Discovery & Exercises (with Angle Bisector Questions)
    1. AAP - Problem 1 (Discovery)
    2. AAP - Problem 2
    3. AAP - Problem 3
    4. Angle Bisector (Problem 1)
  5. Vertical Angles: Theorem Discovery & Problems
    1. Vertical Angles Exploration (1)
    2. Vertical Angles Exploration (2)
    3. Vertical Angles Theorem
    4. VAT (Problem 1)
  6. Course Intro Problem
    1. Where is the Treasure?

1.

Interactive Definition Prompts

  • 1. Collinear Points (Definition Prompt)
  • 2. Parallel Lines Definition
  • 3. Congruent Segments: Quick Exploration
  • 4. Midpoint Definition
  • 5. Congruent Angles: Definition
  • 6. Angle Bisector Definition (I)
  • 7. Perpendicular Lines Definition (& Consequence)
  • 8. Segment Bisectors (Definitions)
  • 9. Perpendicular Bisector Definition
  • 10. Complementary Meaning?
  • 11. Supplementary Angles (Quick Exploration)
  • 12. Triangle Altitude Illustrator and Definition Writing Prompt
  • 13. Triangle Median Illustrator & Definition Writing Prompt

1.1.

Collinear Points (Definition Prompt)

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.
– Tim Brzezinski

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1.11.

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1.13.

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2.

Practice Exercises with Definitions

  • 1. Proving Segments Congruent (Exercise)
  • 2. Proving Angles Congruent (1)
  • 3. Proving Angles Congruent (2)
  • 4. Proving Angles Congruent

2.1.

Proving Segments Congruent (Exercise)

In the applet below, use the provided transformational geometry tools within the limited toolbar [b]to prove [/b]that [color=#9900ff][b]segment f[/b][/color] [b]IS CONGRUENT TO[/b] [color=#444444][b]segment g[/b][/color]. [br][br]Feel free to place the segments wherever you'd like prior to starting this exercise. [br][br]If you don't recall what it means for segments to be considered congruent, [url=https://www.geogebra.org/m/dFADRr9G]click here[/url] for a refresher.
– Tim Brzezinski

2.2.

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3.

Segment Addition Postulate Discovery & Exercises (with Midpoint Questions)

  • 1. SAP Problem 1 (Discovery)
  • 2. SAP - Problem 2
  • 3. SAP - Problem 3
  • 4. Midpoint Definition Question (I)
  • 5. Midpoint Definition Question (II)

3.1.

SAP Problem 1 (Discovery)

The questions you need to answer will appear in the applet below. So get started!
– Tim Brzezinski

3.2.

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3.3.

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3.5.

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4.

Angle Addition Postulate Discovery & Exercises (with Angle Bisector Questions)

  • 1. AAP - Problem 1 (Discovery)
  • 2. AAP - Problem 2
  • 3. AAP - Problem 3
  • 4. Angle Bisector (Problem 1)

4.1.

AAP - Problem 1 (Discovery)

The questions you need to answer will soon appear in the applet below.[br]So, get started!
– Tim Brzezinski

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5.

Vertical Angles: Theorem Discovery & Problems

  • 1. Vertical Angles Exploration (1)
  • 2. Vertical Angles Exploration (2)
  • 3. Vertical Angles Theorem
  • 4. VAT (Problem 1)

5.1.

Vertical Angles Exploration (1)

Interact with this app for a few minutes. Be sure to drag the LARGE POINT around and be sure to move the 2 sliders you see.
Notice anything interesting? Describe as best as you can.
What happens if you make the size of the [b][color=#ff00ff]angle[/color][/b] BIGGER or smaller? [br][br][b][color=#ff00ff](You can change the size of the manila angle by using the manila-colored slider). [/color][/b]Does your response to the question above change? Why? Why not? Describe.
– Tim Brzezinski

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6.

Course Intro Problem

  • 1. Where is the Treasure?

6.1.

Where is the Treasure?

You are on a treasure hunt. On the map below, there is a treasure. The treasure is supposedly buried in the field on which you and your friend are standing. [br]Inside this field are a birdbath [color=#c51414]fountain[/color], [color=#1551b5]flagpole[/color], and a [color=#0a971e]tree[/color] (see applet below.) [b]Here's what you DO know:[/b][br][br]1) The location of the treasure is located at a point that is 15 m away from the [color=#1551b5]flagpole[/color].[br]2) The treasure is just as far away from the [color=#0a971e]tree[/color] as it is from the [color=#c51414]fountain[/color]. [br][br]For this applet below, assume that 1 unit represents 2 m (in real life). Use the tools of GeoGebra in the applet below to determine a possible location of this treasure.[br]After locating a location, label its point "T" (for "Treasure").
Additional Questions:[br][br]1) How many possibilities are there for the location of the treasure? [br]2) If you only found one possible location, find the other possible location of the treasure. [br][br][b]Be sure to use the tools of GeoGebra to CLEARLY DEMONSTRATE/SHOW that the 2 criteria (listed above the applet) are met! [/b]
– Tim Brzezinski

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