Triangle Similarity Theorems Intro
Thank you to Tim Brzezinski for his excellent work on dynamic CCSS materials for geometry. [b]This GeoGebra Book contains lots of discovery-based learning activities, investigations, and meaningful remediation worksheets that were designed to help enhance students' learning of geometry concepts both inside and outside of the mathematics classroom. [color=#1551b5]This GeoGebra Book also contains contributions from the following individuals: Jennifer Silverman ([url]https://www.geogebra.org/jennifer+silverman[/url]) Steve Phelps ([url]https://www.geogebra.org/stevephelps[/url]) Dr. Ted Coe ([url]https://www.geogebra.org/tedcoe[/url]) Without their amazing talent and creativity, this resource would not have been possible. [/b][/color]
Table of Contents
- HSG.SRT.A.2
- Angles in Dilation
- HSG.SRT.A.3
- AA Similarity Theorem
- Animation 97
- AAA: Similar Quads?
- HSG.SRT.B.4 & B.5
- SAS ~ Theorem
- SSS ~ Theorem (V1)
- Animation 63
- SSS ~ Theorem (V2)
- SSSS: Similar Quads?
- Similar Right Triangles (V1)
- Animation 13
- Pythagorean and similar triangles.
- Effortless Trisections
- Animation 134
- Parallel Lines Congruent Segments Theorem
- Parallel Lines Proportionality Theorem
- Triangle-Angle Bisector Theorem
- Animation 81
- Proof Without Words
- Wordless Pythagorean Theorem based on Translations
- Proof Without Words
- Proof Without Words
- Proof Without Words
- Proof Without Words
- Proof Without Words
- Proof Without Words
- Proof Without Words
- Proof Without Words
- Proof Without Words
Angles in Dilation
[color=#000000][b]The following exercises will introduce a new set of triangle similarity theorems that we have been working towards for the past two weeks. Answer the questions as well as you can, and pay attention to all new triangle similarity theorems you encounter. We will be adding these to our special notebooks today.[/b][br]___________________________[br][br]Interact with the applet below for a few minutes, then answer the following question.[br][br][/color][color=#0000ff][b][i]What is the effect of a dilation on the angles of a triangle? [/i][/b][/color]
AA Similarity Theorem
[color=#000000]The [/color][b][color=#0000ff]AA Similarity Theorem[/color][/b][color=#000000] states:[/color][br][br][i][color=#0000ff]If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. [/color][/i][br][br][color=#980000]Below is a visual that was designed to help you prove this theorem true in the case where both triangles have the same orientation. (If the triangles had opposite orientations, you would have to first [b]reflect[/b] the white triangle [b]about any one of its sides[/b] first, and then proceed along with the steps taken in the applet.) [/color][br][br][color=#000000]Feel free to move the locations of the [/color][color=#38761d][b]BIG GREN VERTICES[/b][/color][color=#000000] of either triangle before slowly dragging the slider. [/color][b] [/b][i][color=#ff0000]Pay careful attention to what happens as you do.[/color][/i]
Quick (Silent) Demo
SAS ~ Theorem
[color=#000000]In the applet below, you'll find two triangles. [br][br]The [b]black angle[/b] in the [/color][color=#38761d][b]green triangle[/b][/color] [b][color=#000000]is congruent to[/color][/b][color=#000000] the [/color][b][color=#000000]black angle[/color][/b][color=#000000] in the [/color][b][color=#ff00ff]pink triangle[/color][/b][color=#000000]. [/color][br][br][color=#000000]In the [/color][color=#38761d][b]green triangle[/b][/color][color=#000000], the [b]black angle is the included angle between sides [/b][/color][b][i][color=#000000]a[/color][/i][color=#000000] and [/color][i][color=#000000]b[/color][/i][/b][color=#000000]. [/color][br][color=#000000]In the [/color][b][color=#ff00ff]pink triangle[/color][/b][color=#000000], the [b]black angle is the included angle between sides [i]ka[/i] and [i]kb[/i][/b]. [/color][br][br][color=#000000]Interact with the applet below for a few minutes. [/color][color=#000000]As you do, be sure to move the locations of the [/color][color=#38761d][b]green triangle's[/b][/color][color=#000000] [b]BIG BLACK VERTICES[/b] and the location of the [b]BIG X[/b].[br][/color][color=#000000]You can also adjust the value of [/color][i][color=#000000]k[/color][/i][color=#000000] by using the slider or by entering a value between 0 & 1. [/color][color=#000000] [br][/color][color=#000000] [/color][br]
[color=#000000]Notice how these two triangles have 2 pairs of corresponding sides that are in proportion. (After all, as long as [br]a > 0 & b > 0, ka/a = k and kb/b = k, right? ) [br][br]The [b]BLACK ANGLES INCLUDED[/b] between these two sides [b]ARE CONGRUENT[/b] as well. [/color][br][br][b][color=#0000ff]From your observations, what can you conclude about the two triangles? Why can you conclude this?[br]Clearly justify your response! [/color][/b]