Unit 4: Parallel and Perpendicular…

Jamie Nordstrom, Aug 16, 2017

Table of Contents

  1. 3.1 Lines and Angles
    1. Transversal Intersects Parallel Lines
    2. Naming Angle Positions
    3. Parallel, Perpendicular, and Skew Lines
    4. Skew Lines
    5. Explore! Parallel Lines Intersected by a Transversal
  2. 3.2 Angles Formed by Parallel Lines and Transversals
    1. Corresponding Angles: Quick Investigation
    2. Congruent Corresponding Angles to Start? (Quick Investigation)
    3. Exploring Corresponding Angles (V2)
    4. Alternate Interior Angles: Quick Investigation
    5. Alternate Interior Angles Theorem (V1)
    6. Exploring Alternate Interior Angles (V2)
    7. Alternate Exterior Angles: Quick Investigation
    8. Alternate Exterior Angles Theorem (V1)
    9. Exploring Alternate Exterior Angles (V2)
    10. Same-Side-Interior Angles: Quick Investigation
    11. Same Side Interior Angles Theorem
    12. Same-Side-Exterior Angles: Quick Investigation
    13. Exploring Same Side Exterior Angles
    14. Transversal Intersects Parallel Lines
  3. 3.3 Proving Lines Parallel
    1. Corresponding Angles Converse Postulate
    2. Consecutive Interior Angles Converse Theorem
    3. Alternate Interior Angles Converse Theorem
    4. Example 1
    5. Alternate Exterior Angles Converse Theorem
    6. Example 3
    7. Example 2
    8. Example 4
  4. 3.4 Perpendicular Lines
  5. 3.5 Slopes of Lines
    1. Slope: Intuitive Introduction
    2. Parallel Consequence? (VA)
  6. 3.6 Lines in the Coordinate Plane
    1. Parallel & Perpendicular Consequence
    2. Perpendicular Lines Investigation (VB)
  7. tasks for assessing and extending 3.1-3.4
    1. Parallel Lines Congruent Segments Theorem
    2. Animation 16
    3. Animation 39
    4. Animation 60
    5. Animation 29
    6. Animation 73
    7. Solution to Parallel Lines & Angles Problem
    8. Effortless Trisections
    9. Animation 134
    10. Parallel Lines Proportionality Theorem

1.

3.1 Lines and Angles

  • 1. Transversal Intersects Parallel Lines
  • 2. Naming Angle Positions
  • 3. Parallel, Perpendicular, and Skew Lines
  • 4. Skew Lines
  • 5. Explore! Parallel Lines Intersected by a Transversal

1.1.

Transversal Intersects Parallel Lines

[b][color=#980000]Students:[/color][/b][br][br]Use the GeoGebra applet above to help you complete the [b][color=#1e84cc]Transversals, Lines, & Related Angles[/color][/b] investigation given to you at the beginning of class.
– Tim Brzezinski

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1.4.

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1.5.

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2.

3.2 Angles Formed by Parallel Lines and Transversals

  • 1. Corresponding Angles: Quick Investigation
  • 2. Congruent Corresponding Angles to Start? (Quick Investigation)
  • 3. Exploring Corresponding Angles (V2)
  • 4. Alternate Interior Angles: Quick Investigation
  • 5. Alternate Interior Angles Theorem (V1)
  • 6. Exploring Alternate Interior Angles (V2)
  • 7. Alternate Exterior Angles: Quick Investigation
  • 8. Alternate Exterior Angles Theorem (V1)
  • 9. Exploring Alternate Exterior Angles (V2)
  • 10. Same-Side-Interior Angles: Quick Investigation
  • 11. Same Side Interior Angles Theorem
  • 12. Same-Side-Exterior Angles: Quick Investigation
  • 13. Exploring Same Side Exterior Angles
  • 14. Transversal Intersects Parallel Lines

2.1.

Corresponding Angles: Quick Investigation

In the applet below, transversal [i]n[/i] intersects the lines [i]g[/i] and [i]i[/i]. [br][br]Interact with the applet below for a few minutes. [br]Then answer the questions that follow.
1.
Use the [b]Slope[/b] tool to measure the slopes of the two lines [i]g[/i] and [i]i[/i]. What do you notice?
2.
What does your observation from (1) tell you about lines [i]g[/i] and [i]i[/i]?
3.
If you haven't done so yet, drag the white point along transversal [i]n[/i] as far as it will go. [color=#9900ff][b]What geometric transformation is taking place?[/b][/color]
4.
If a transversal (in this case, [i]n[/i]) intersects 2 lines that have the property that you wrote in response to (2) above, what can you conclude about the pair of [color=#ff7700][b]orange corresponding angles [/b][/color]that were formed?
– Tim Brzezinski

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2.8.

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2.10.

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2.11.

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2.12.

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2.13.

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2.14.

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3.

3.3 Proving Lines Parallel

  • 1. Corresponding Angles Converse Postulate
  • 2. Consecutive Interior Angles Converse Theorem
  • 3. Alternate Interior Angles Converse Theorem
  • 4. Example 1
  • 5. Alternate Exterior Angles Converse Theorem
  • 6. Example 3
  • 7. Example 2
  • 8. Example 4

3.1.

Corresponding Angles Converse Postulate

[size=150][b]Corresponding [/b][b]Angles Converse Postulate[br][br][/b]If the corresponding angles are ________________ then the lines are parallel.[/size][br]
– kherdegen

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3.5.

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3.6.

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3.7.

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3.8.

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4.

3.4 Perpendicular Lines


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5.

3.5 Slopes of Lines

  • 1. Slope: Intuitive Introduction
  • 2. Parallel Consequence? (VA)

5.1.

Slope: Intuitive Introduction

[color=#000000]Discovery Lesson Activity: [br][/color][color=#980000][br][/color][b][color=#0000ff][url=https://docs.google.com/document/d/1W5fggl-QmnwOoT1Yjs2MfZnwVzIF49pflGX33xe0PN0/edit?usp=sharing]Slope: Intuitive Introduction Investigation [/url][/color][/b]
– Tim Brzezinski

5.2.

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6.

3.6 Lines in the Coordinate Plane

  • 1. Parallel & Perpendicular Consequence
  • 2. Perpendicular Lines Investigation (VB)

6.1.

Parallel & Perpendicular Consequence

[color=#000000]The following applet demonstrates a property that parallel lines have when they're drawn in the coordinate plane. [br] [i] [br]Be sure to move the [color=#1e84cc][b]blue points[/b][/color] around quite a bit![/i] [/color]
[color=#000000]This applet demonstrates a property that perpendicular lines have when they're drawn in the coordinate plane. [br][br][i]Be sure to move the points around quite a bit and observe carefully as you do! [/i][/color]
[color=#000000]What can you conclude about parallel lines drawn in the coordinate plane? [/color]
[color=#000000]What can you conclude about perpendicular lines that are drawn in the coordinate plane? [br](Assume the lines are not aligned horizontally and vertically). [/color]
– Tim Brzezinski

6.2.

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7.

tasks for assessing and extending 3.1-3.4

  • 1. Parallel Lines Congruent Segments Theorem
  • 2. Animation 16
  • 3. Animation 39
  • 4. Animation 60
  • 5. Animation 29
  • 6. Animation 73
  • 7. Solution to Parallel Lines & Angles Problem
  • 8. Effortless Trisections
  • 9. Animation 134
  • 10. Parallel Lines Proportionality Theorem

7.1.

Parallel Lines Congruent Segments Theorem

Interact with the following applet for a few minutes.  Be sure to change the location of each BIG point.  Be sure to move the slider at some point as well.  Then complete the activity that follows.  
The applet above suggests a very powerful, yet often unused theorem in all of geometry. See if you can rearrange the list of words below to form a conditional ("If-then") statement that states what is shown in the applet above!  [br][br][color=#0000ff]segments        one           parallel             then[br][br]every      cut        transversal    segments      lines[br][br]or       congruent      they       transversal     cut[br][br]If      congruent     off       on       more[br][br]3   on    off[br][/color]
Write a formal proof of the conditional statement (whose words you've arranged) above.  
– Tim Brzezinski

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7.10.

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