Geometry: Introductory Definitions, Postulates, Theorems

Tim Brzezinski, Jenny Secor, Jun 7, 2016

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. Perpendicular Bisector Definition
    11. Complementary Meaning?
    12. Supplementary Angles (Quick Exploration)
  2. Pythagorean Theorem Interactive Discovery
    1. What Do You See Here?
  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. VAT (Problem 1)
    2. Vertical Angles Theorem
  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. Perpendicular Bisector Definition
  • 11. Complementary Meaning?
  • 12. Supplementary Angles (Quick Exploration)

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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1.5.

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1.6.

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1.7.

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1.8.

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1.9.

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1.10.

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1.11.

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1.12.

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2.

Pythagorean Theorem Interactive Discovery

  • 1. What Do You See Here?

2.1.

What Do You See Here?

In the applet below, you'll notice a [b]RIGHT TRIANGLE[/b] (shown in white) with colorful squares built off its 2 legs. There is a blank (white) square built off its hypotenuse. [br][br]1) Drag the gray slider all the way to the right. [br]2) What do you notice? Describe in detail! [br][br]3) Now, drag the slider all the way back to the left side.[br]4) Move the vertices (corners) of the right triangle around so that the right triangle doesn't stay the same as the right triangle you started with. [br]5) Drag the gray slider all the way down to the right. What do you notice? Describe in detail![br][br]6) Repeat steps (3) - (5) as many times as you wish? [br][br]7) [b]Can you make a general statement about the area of the square built off the shorter leg, [br] the area of the square built off the longer leg, and the area of the square built off the hypotenuse? [/b][br] [b] [color=#c51414]Write this statement in a complete sentence, and be prepared to share this statement with the rest of the class! [/color][/b]
What Do You See Here?
– Tim Brzezinski, Jed Butler

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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3.4.

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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

4.2.

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4.3.

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4.4.

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5.

Vertical Angles: Theorem Discovery & Problems

  • 1. VAT (Problem 1)
  • 2. Vertical Angles Theorem

5.1.

VAT (Problem 1)

Using what you've just recently discovered about vertical angles to solve the problem below (see applet).
– Tim Brzezinski

5.2.

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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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