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CCSS High School: Geometry (Similarity, Right T, Trig.)

Tim Brzezinski, John Williams, Nov 29, 2016

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. This GeoGebra Book also contains contributions from the following individuals: Jennifer Silverman (https://www.geogebra.org/jennifer+silverman) Steve Phelps (https://www.geogebra.org/stevephelps) Dr. Ted Coe (https://www.geogebra.org/tedcoe) Without their amazing talent and creativity, this resource would not have been possible.

HSG.SRT.A.1, A.1.A, A.1.B
1. Dilation Introduction: Transformers!
2. Dilating a Point (Intro)
3. Dilating a Line
4. Dilating a Segment:
HSG.SRT.A.2
1. Properties of Dilations
2. Properties of Dilations (II)
HSG.SRT.A.3
1. AA Similarity Theorem
2. Animation 97
3. AAA: Similar Quads?
HSG.SRT.B.4 & B.5
1. SAS ~ Theorem
2. SSS ~ Theorem (V1)
3. Animation 63
4. SSS ~ Theorem (V2)
5. SSSS: Similar Quads?
6. Similar Right Triangles (V1)
7. Animation 13
8. Pythagorean and similar triangles.
9. Effortless Trisections
10. Animation 134
11. Parallel Lines Congruent Segments Theorem
12. Parallel Lines Proportionality Theorem
13. Triangle-Angle Bisector Theorem
14. Animation 81
15. Proof Without Words
16. Wordless Pythagorean Theorem based on Translations
17. Proof Without Words
18. Proof Without Words
19. Proof Without Words
20. Proof Without Words
21. Proof Without Words
22. Proof Without Words
23. Proof Without Words
24. Proof Without Words
25. Proof Without Words
HSG.SRT.C.6 & C.7 & C.8
1. Right Triangle Trigonometry: Intro
2. Discovering Sine, Cosine, and Tangent in Right Triangles
3. Trigonometric Ratios (Right Triangle Context)
4. What "CO" in COsine Means
5. Animation 98
6. Animation 99
7. Animation 100
8. How Fast are You Spinning?
HSG.SRT.D.9 & D.10 & D.11
1. Law of Sines (& Area)
2. Law of Sines (Illustrated)
3. Law of Cosines: Discovery
Links to Other CCSS High School: Geometry Standards
1. Geometry Resources

Information

1.

HSG.SRT.A.1, A.1.A, A.1.B

1. Dilation Introduction: Transformers!
2. Dilating a Point (Intro)
3. Dilating a Line
4. Dilating a Segment:

1.1.

Dilation Introduction: Transformers!

Explore this applet to see how the center of dilation and the scale factor affect the image.

Turn off the images and explore the way the lines change. What do you notice?

1.2.

1.3.

1.4.

2.

HSG.SRT.A.2

1. Properties of Dilations
2. Properties of Dilations (II)

2.1.

Properties of Dilations

This applet accompanies the Introduction to Dilations activity given to you in class. Have fun with this!

2.2.

3.

HSG.SRT.A.3

1. AA Similarity Theorem
2. Animation 97
3. AAA: Similar Quads?

3.1.

AA Similarity Theorem

The AA Similarity Theorem states: If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. 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 reflect the white triangle about any one of its sides first, and then proceed along with the steps taken in the applet.) Feel free to move the locations of the BIG GREN VERTICES of either triangle before slowly dragging the slider. Pay careful attention to what happens as you do.

Quick (Silent) Demo

3.2.

3.3.

4.

HSG.SRT.B.4 & B.5

1. SAS ~ Theorem
2. SSS ~ Theorem (V1)
3. Animation 63
4. SSS ~ Theorem (V2)
5. SSSS: Similar Quads?
6. Similar Right Triangles (V1)
7. Animation 13
8. Pythagorean and similar triangles.
9. Effortless Trisections
10. Animation 134
11. Parallel Lines Congruent Segments Theorem
12. Parallel Lines Proportionality Theorem
13. Triangle-Angle Bisector Theorem
14. Animation 81
15. Proof Without Words
16. Wordless Pythagorean Theorem based on Translations
17. Proof Without Words
18. Proof Without Words
19. Proof Without Words
20. Proof Without Words
21. Proof Without Words
22. Proof Without Words
23. Proof Without Words
24. Proof Without Words
25. Proof Without Words

4.1.

SAS ~ Theorem

In the applet below, you'll find two triangles. The black angle in the green triangle is congruent to the black angle in the pink triangle. In the green triangle, the black angle is the included angle between sides a and b. In the pink triangle, the black angle is the included angle between sides ka and kb. Interact with the applet below for a few minutes. As you do, be sure to move the locations of the green triangle's BIG BLACK VERTICES and the location of the BIG X. You can also adjust the value of k by using the slider or by entering a value between 0 & 1.

Notice how these two triangles have 2 pairs of corresponding sides that are in proportion. (After all, as long as a > 0 & b > 0, ka/a = k and kb/b = k, right? ) The BLACK ANGLES INCLUDED between these two sides ARE CONGRUENT as well. From your observations, what can you conclude about the two triangles? Why can you conclude this? Clearly justify your response!

4.2.

4.3.

4.4.

4.5.

4.6.

4.7.

4.8.

4.9.

4.10.

4.11.

4.12.

4.13.

4.14.

4.15.

4.16.

4.17.

4.18.

4.19.

4.20.

4.21.

4.22.

4.23.

4.24.

4.25.

5.

HSG.SRT.C.6 & C.7 & C.8

1. Right Triangle Trigonometry: Intro
2. Discovering Sine, Cosine, and Tangent in Right Triangles
3. Trigonometric Ratios (Right Triangle Context)
4. What "CO" in COsine Means
5. Animation 98
6. Animation 99
7. Animation 100
8. How Fast are You Spinning?

5.1.

Right Triangle Trigonometry: Intro

This applet accompanies the Right Triangle Trigonometry: Intro activity packet you received at the beginning of class today. Use this applet to help you complete the guiding questions in this activity.

Key Question: 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)! Why does this occur? Explain!

5.2.

5.3.

5.4.

5.5.

5.6.

5.7.

5.8.

6.

HSG.SRT.D.9 & D.10 & D.11

1. Law of Sines (& Area)
2. Law of Sines (Illustrated)
3. Law of Cosines: Discovery

6.1.

Law of Sines (& Area)

Interact with the applet below for a minute. Then, answer the questions that follow. (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. Write an equation that expresses the relationship among angle B, the triangle's height, and side c.

2) Rewrite this equation so that height is written in terms of side c and angle B.

3) Now consider the pink right triangle on the right. Write an equation that expresses the relationship among angle C, side b, and the triangle's height.

4) Rewrite this equation so that height is written in terms of side b and angle C.

5) Take your responses to questions (2) and (4) to write a new equation that expresses the relationship among C, B, c, and b. Write this equation so that C and c appear on one side of the equation and that B and b appear on the other.

6) Now drag the slider in the upper right hand corner. Now, given the fact that the length of segment BC would be denoted as a (it's just not drawn in the applet above), write an expression for the area of this original triangle in terms of a, b, and C.

7) Same question as in (6) above, but this time write the area of the triangle in terms of a, c, and B.

8) Suppose that dragging the first slider dropped a height from point C instead of point A. Answer questions (1) - (5) again, this time letting c serve as the base of this triangle (vs. side a). Notice anything interesting in your results?

Links to Other CCSS High School: Geometry Standards

1. Geometry Resources

7.1.

Geometry Resources

Congruence (Volume 1)
Congruence (Volume 2)
Similarity, Right Triangles, Trigonometry
Circles
Coordinate and Analytic Geometry
Area, Surface Area, Volume, 3D, Cross Section 
Proof Challenges

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