CCSS High School: Geometry (Similarity, Right T, Trig.)
[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
- Properties of Dilations
- Properties of Dilations (II)
- 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
- HSG.SRT.C.6 & C.7 & C.8
- Right Triangle Trigonometry: Intro
- Discovering Sine, Cosine, and Tangent in Right Triangles
- Trigonometric Ratios (Right Triangle Context)
- What "CO" in COsine Means
- Animation 98
- Animation 99
- Animation 100
- How Fast are You Spinning?
- HSG.SRT.D.9 & D.10 & D.11
- Law of Sines (& Area)
- Law of Sines (Illustrated)
- Law of Cosines: Discovery
- Links to Other CCSS High School: Geometry Standards
- Geometry Resources
Properties of Dilations
This applet accompanies the [b]Introduction to Dilations[/b] activity given to you in class. [br]Have fun with this!
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]
Right Triangle Trigonometry: Intro
[color=#000000]This applet accompanies the [/color][i][color=#0000ff]Right Triangle Trigonometry: Intro[/color][/i][color=#000000] activity packet you received at the beginning of class today. Use this applet to help you complete the guiding questions in this activity. [/color]
[color=#980000][b]Key Question: [/b][/color] [br][br][color=#000000]Hopefully, you noticed that when you keep the blue angle fixed and move only the white vertices, the value of each ratio never changes (even though the side lengths do)! [/color][color=#ff00ff][i]Why does this occur? [/i] Explain! [/color]
Law of Sines (& Area)
Interact with the applet below for a minute. [br]Then, answer the questions that follow. [br](Please don't slide the 2nd slider until prompted to in the directions below.)
1) Take a look at the yellow right triangle on the left.[br]Write an equation that expresses the relationship among angle [i]B[/i], the triangle's height, and side [i]c[/i].
2) Rewrite this equation so that [i]height [/i]is written in terms of side [i]c[/i] and angle [i]B[/i].
3) Now consider the pink right triangle on the right. Write an equation that expresses the relationship among angle [i]C[/i], side [i]b[/i], and the triangle's height. [br]
4) Rewrite this equation so that [i]height[/i] is written in terms of side [i]b[/i] and angle [i]C[/i].
5) Take your responses to questions (2) and (4) to write a new equation that expresses the relationship among [i]C[/i], [i]B[/i], [i]c[/i], and [i]b[/i]. Write this equation so that [i]C[/i] and [i]c[/i] appear on one side of the equation and that [i]B[/i] and [i]b[/i] appear on the other.
6) Now drag the slider in the upper right hand corner. Now, given the fact that the length of segment [i]BC[/i] would be denoted as [i]a [/i](it's just not drawn in the applet above), write an expression for the area of this original triangle in terms of [i]a[/i], [i]b[/i], and [i]C[/i].
7) Same question as in (6) above, but this time write the area of the triangle in terms of [i]a[/i], [i]c[/i], and [i]B[/i].
8) Suppose that dragging the first slider dropped a height from point [i]C[/i] instead of point [i]A[/i]. Answer questions (1) - (5) again, this time letting [i]c[/i] serve as the base of this triangle (vs. side [i]a[/i]). Notice anything interesting in your results?
Geometry Resources
[list][*][b][url=https://www.geogebra.org/m/z8nvD94T]Congruence (Volume 1)[/url][/b][/*][*][b][url=https://www.geogebra.org/m/munhXmzx]Congruence (Volume 2)[/url][/b][/*][*][b][url=https://www.geogebra.org/m/dPqv8ACE]Similarity, Right Triangles, Trigonometry[/url][/b][/*][*][b][url=https://www.geogebra.org/m/C7dutQHh]Circles[/url][/b][/*][*][b][url=https://www.geogebra.org/m/K2YbdFk8]Coordinate and Analytic Geometry[/url][/b][/*][*][b][url=https://www.geogebra.org/m/xDNjSjEK]Area, Surface Area, Volume, 3D, Cross Section[/url] [/b][/*][*][b][url=https://www.geogebra.org/m/NjmEPs3t]Proof Challenges[/url] [/b][/*][/list]