Proving Triangles Congruent
Contains applets in which students prove two triangles congruent using the tools of transformational geometry.
Table of Contents
- Proving Triangles Congruent using Transformations
- SAS - Exercise 1A
- SAS - Exercise 1B
- SAS - Exercise 2
- SAS - Exercise 3
- ASA: Exercise
- Proving Triangles Congruent (Dynamic Proofs)
- SSS: Dynamically Illustrated
- SAS: Dynamic Proof!
- SSS: Dynamic Proof!
- ASA Theorem?
- Proving Triangles Congruent using Coordinate Geometry
- Slope to Angle Measure Calculator (I)
- Slope to Angle Measure Calculator (II)
- SSA ???
- SSA Theorem?
- Ambiguous Case (SSA) Illustrator: All Cases
SAS - Exercise 1A
In the applet below, use the tools of transformational geometry to informally demonstrate the SAS Triangle Theorem to be true. [br][br]That is, use the tools of transformational geometry to map the [color=#bf9000][b]yellow triangle[/b][/color] onto the empty triangle. [br][br]Before starting, feel free to adjust any aspect of the [color=#bf9000][b]starting triangle[/b][/color] ([color=#666666][b]tilt[/b][/color], [color=#1e84cc][b]size of the included angle[/b][/color], [b]and the positions of points [/b][i]A[/i][b], [/b][i]B[/i][b], and [/b][i]C[/i]). You can also use the [b]black slider[/b] to [b]change the position of the image (empty) triangle.[/b] [br][br][i]Once you do start, it is recommended that you don't readjust these parameters. [/i]
Question:
Describe how you know the two triangle are congruent.
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]
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]
SSA Theorem?
[b]Thus far, we've learned several theorems that allow us to conclude 2 triangles are congruent. [br][br]Here's the list of discoveries we've made thus far: [br][/b][br][url=https://www.geogebra.org/m/bM5FkyFK]SAS Theorem[/url][br][url=https://www.geogebra.org/m/Qsk3vDs6]SSS Theorem[/url][br][url=https://www.geogebra.org/m/WKJJ2uPa]ASA Theorem[/url][br]AAS Theorem (easily proven simply by finding the each triangle's 3rd angle and then using ASA Theorem.)[br][br]HL Theorem (For Right Triangles: Easily Proven since we can just use the Pythagorean Theorem to solve for the other leg and then use the SSS Theorem.) [br][br][b][color=#0000ff]Yet MANY students ask, "What about SSA?" [br][br][/color][color=#0000ff]That is, if 2 sides and a non-included-angle of one triangle are congruent to 2 sides and a non-included-angle of another triangle, are the triangles themselves congruent? [br][br][/color]Interact with BOTH applets for a few minutes and see if you can answer this question for yourself. [/b][i][color=#9900ff]As you do, feel free to move the WHITE POINTS anywhere you'd like! [br]Feel free to adjust the "a" and "b" sliders as well. [br][/color][/i]