Proving Triangles Congruent

Tim Brzezinski, Tim Ladwig, Mar 8, 2017

Contains applets in which students prove two triangles congruent using the tools of transformational geometry.

Table of Contents

  1. Proving Triangles Congruent using Transformations
    1. Proving Tri's Congruent (I)
    2. Proving Tri's Congruent (II)
    3. SAS - Exercise 1A
    4. SAS - Exercise 1B
    5. SAS - Exercise 2
    6. SAS - Exercise 3
    7. SSS: Exercise 1
    8. SSS: Exercise 2
    9. SSS: Exercise 3
  2. Proving Triangles Congruent (Dynamic Proofs)
    1. SSS: Dynamically Illustrated
    2. SAS: Dynamic Proof!
    3. SSS: Dynamic Proof!
    4. ASA Theorem?
  3. Proving Triangles Congruent using Coordinate Geometry
    1. Slope to Angle Measure Calculator (I)
    2. Slope to Angle Measure Calculator (II)
  4. SSA ???
    1. Is "SSA" Legit? What Do You Think?

1.

Proving Triangles Congruent using Transformations

  • 1. Proving Tri's Congruent (I)
  • 2. Proving Tri's Congruent (II)
  • 3. SAS - Exercise 1A
  • 4. SAS - Exercise 1B
  • 5. SAS - Exercise 2
  • 6. SAS - Exercise 3
  • 7. SSS: Exercise 1
  • 8. SSS: Exercise 2
  • 9. SSS: Exercise 3

1.1.

Proving Tri's Congruent (I)

[color=#000000]Recall an [b]isometry[/b] is a [b]transformation that preserves distance. [br][/b][br]Also recall that, by definition, 2 polygons are said to be congruent polygons if and only if one polygon can be mapped perfectly onto the other polygon using an isometry or a composition of two or more isometries. [br][br]Use the tools of GeoGebra to show, by definition, that the following two triangles are congruent. [/color]
– Tim Brzezinski

1.2.

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1.3.

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1.4.

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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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2.

Proving Triangles Congruent (Dynamic Proofs)

  • 1. SSS: Dynamically Illustrated
  • 2. SAS: Dynamic Proof!
  • 3. SSS: Dynamic Proof!
  • 4. ASA Theorem?

2.1.

SSS: Dynamically Illustrated

[color=#000000]Feel free to move the [b]BIG WHITE POINTS[/b] anywhere you'd like![br]You can adjust the triangle's side lengths by using the vertical sliders on the right. [/color]
– Tim Brzezinski

2.2.

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2.3.

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2.4.

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3.

Proving Triangles Congruent using Coordinate Geometry

  • 1. Slope to Angle Measure Calculator (I)
  • 2. Slope to Angle Measure Calculator (II)

3.1.

Slope to Angle Measure Calculator (I)

[b]Students: [color=#1551b5]This applet is to be used when trying to use coordinate geometry prove two triangles (drawn in the coordinate plane) congruent.[/color] [/b][br][br][color=#c51414][i]You already know how to calculate the distance between any two points in the coordinate plane, so finding the length of any side of a triangle should be an easy task. [br]However, calculating the measure of an interior angle between two sides of a triangle is a bit more complicated.[/i][/color][br][br][color=#1551b5][b]If you need to calculate the angle measure between any two sides of a triangle, follow the directions in the applet below: [/b] [/color]
– Tim Brzezinski

3.2.

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4.

SSA ???

  • 1. Is "SSA" Legit? What Do You Think?

4.1.

Is "SSA" Legit? What Do You Think?

[color=#274e13][i]So far, we've learned a few theorems that help us prove two triangles congruent.  Recall these theorems are:[/i][/color][br][br][b][color=#0000ff]The SSS Theorem[br]The SAS Theorem[br]The ASA Theorem[/color][/b][br][br]Yet, what about [color=#ff0000][b]"SSA"[/b][/color]?  That is, [b][color=#ff0000]if we have 2 sides and a "non-included" angle of one triangle congruent to 2 sides and a "non-included" angle of another triangle, can we conclude that the two triangles are congruent[/color][/b]?  [br][br]Interact with the applet below for a minute or two.  (Feel free to move any of the big black points around.)  Then answer the [color=#ff0000][b]question[/b][/color] above (in detail.)  
– Tim Brzezinski

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