Triangle Existence

Harmony Richman, 1 mars 2026

Discover the Triangle Inequality Theorem and Angle Sum Theorem through construction. Differentiate between conditions that result in a Unique triangle, No triangle, or Infinitely Many triangles. Use GeoGebra tools to prove geometric constraints.

Table des matières

  1. Triangle Inequality Theorem
    1. Triangle Inequality Theorem
  2. Triangle Angle Sum Theorem
    1. Triangle Sum Exploration
  3. Triangle Investigation Tasks
    1. Triangle Task #1
    2. Triangle Task #2
    3. Triangle Task #3
    4. Triangle Task #4
    5. Triangle Task #5
    6. Triangle Task #6
    7. Triangle Task #7
    8. Triangle Task #8
    9. Triangle Task #9
    10. Triangle Task #10
  4. Reflection
    1. Reflection

1.

Triangle Inequality Theorem

9-10.GM.10 Prove and apply theorems about triangles.

  • 1. Triangle Inequality Theorem

1.1.

Triangle Inequality Theorem

Look at the construction below. Can you move the points in the construction so that segments a, b, and c form a triangle? In this exploration, you will determine the conditions required for side lengths to form triangles. This set of conditions is known as the Triangle Inequality Theorem.
Set the side lengths a, b, and c to 7, 10, and 19, respectively. Can these three segments form a triangle? Why or why not?[br]
Use the slider to adjust the length of side a only. As you do so, manipulate the points to try to form triangles. What is the smallest value of side a that will allow you to create a triangle? What is the largest value of side a that will allow you to create a triangle?[br]
Experiment with the lengths of all three sides. What has to be true about the relationships between the sides in order to be able to form triangles?
[b]Triangle Inequality Theorem[br][/b]For 3 sides of a triangle: A, B, C.[br]The sum of any 2 sides MUST be greater than the 3rd side.[br][br]A + B > C[br]A + C > B[br]B + C > A
Determine whether the side lengths can form a triangle or not?[br][br]11, 4, 18[br]
Determine whether the side lengths can form a triangle or not?[br][br]13, 4, 7
Determine whether the side lengths can form a triangle or not?[br][br]8, 3, 7
– Harmony Richman, Steve Miller

2.

Triangle Angle Sum Theorem

8.GM.GF.4 Describe the following angle-pair relationships: interior and exterior angles of triangles and angles formed when a transversal cuts parallel lines or intersecting lines. Solve for an unknown angle in a figure by applying facts about these angles.

  • 1. Triangle Sum Exploration

2.1.

Triangle Sum Exploration

Draw a Triangle, then measure all three angles of the triangle.
What is the sum of your angles?
Move your points around, then add them up again. Does the sum stay the same?
Write a conjecture (a statement based on evidence) about the angles in any triangle.
This is called the TRIANGLE SUM THEOREM
Test your conjecture by moving the points around in the diagram below.
Triangle Sum Theorem
Let's prove your conjecture. In the diagram below, follow these steps:[br]1. Create a line parallel to Line BC through Point A. (do this by clicking the "parallel line" tool, then clicking side BC then clicking point A).[br]2. Create a point on both sides of Point A. (do this by clicking the "point on object" tool, then clicking on the new line to the left of A, then click on the new line to the right of A).[br]3. Measure the 2 new created angles (Angle DAB and Angle CAE).[br]4. If any angles are congruent, color them the same color.
What kind of angles are Angle DAB and Angle ABC and what is their relationship?
What kind of angles are Angle CAE and Angle BCA and what is their relationship?
What kind of angles are Angle DAB, Angle BAC, and Angle CAE and what is their relationship?
Explain how this activity proves that the Triangle Sum Theorem
– MrCarmack

3.

Triangle Investigation Tasks

You will use the following tasks to test out your paper results. 7.GM.GF.1 Draw triangles from given conditions using appropriate tools. Defend whether a unique triangle, multiple triangles, or no triangle can be constructed when given three measures of angles or sides.

  • 1. Triangle Task #1
  • 2. Triangle Task #2
  • 3. Triangle Task #3
  • 4. Triangle Task #4
  • 5. Triangle Task #5
  • 6. Triangle Task #6
  • 7. Triangle Task #7
  • 8. Triangle Task #8
  • 9. Triangle Task #9
  • 10. Triangle Task #10

3.1.

Triangle Task #1

An acute triangle with one angle that measures 38[math]^\circ[/math][br][br]Are all sides and angles labeled with their measurements?[br][br]In GeoGebra, a construction is considered correct if it remains true to its properties even when you try to drag its vertices.[br][br]If I drag a vertex, does the triangle stay "true" to the original prompt. [br][list][*]If it stays the same size - It's Unique.[/*][*]If it changes size but keeps its shape - It has Many Solutions.[/*][*]If it breaks - It might be Impossible.[/*][/list]Use the silent video below to help get you started with your construction.
– Harmony Richman

3.2.

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3.3.

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3.4.

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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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3.9.

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3.10.

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4.

Reflection

  • 1. Reflection

4.1.

Reflection

Let's take a moment to reflect on the triangle activities.
Would you have your future students use GeoGebra [i]before[/i] drawing by hand to see what is possible, or [i]after[/i] as a way to check their work? Defend your sequence based on today’s experience.
In Triangle Task #8, where you had created a triangle with angles of 40[math]^\circ[/math], 60[math]^\circ[/math], and 80[math]^\circ[/math] GeoGebra allows you to drag a vertex and change the triangle's size while the angles stay the same. How does this visual "stretching" provide a better defense for "Infinitely Many Solutions" than a static paper drawing?
How would you explain the difference between a "Unique" triangle and a "Multiple" triangles to a student who thinks "if I can draw it, it's unique."
For a learner with fine-motor challenges (dysgraphia), how does GeoGebra change the playing field for geometry? Conversely, what is lost if that learner never picks up a physical protractor?
– Harmony Richman

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