 # K12MATH010: Geometry

Unit 2: Similarity, Proof, and Trigonometry   Pilots trying to avoid collisions with mountains, engineers trying to build safe suspension bridges, surveyors trying to assist with city planning, and architects trying to design music halls with rich acoustics all rely on their understanding of similarity and trigonometry to be successful. This unit will also build on your knowledge of transformations, but this time you will look at a nonrigid transformation, the dilation and, in conjunction with proportional reasoning, you will develop a formal understanding of similarity. You will investigate shortcuts for determining similarity between triangles and solve problems involving similar figures with a heavy emphasis on triangles. Next, you will be introduced to triangle trigonometry, and you will learn about special right triangles and relate their properties to the Pythagorean theorem, studied in previous courses. Finally, you will learn to find the missing measures of general triangles, that is to say triangles that are not necessarily right triangles, using the laws of sines and cosines.[](#_ftn1)

Ibid

Completing this unit should take approximately 8 hours.

☐    Subunit 2.1: 2 hours

☐    Subunit 2.2: 1 hour, 30 minutes

☐    Subunit 2.3: 4 hours, 30 minutes

Unit2 Learning Outcomes
Upon successful completion of this unit, you will be able to:
- Explain similarity in terms of similarity transformations. - Prove theorems involving similarity. - Define trigonometric ratios and solve problems involving right triangles. - Apply geometric concepts in modeling situations. - Apply trigonometry to general triangles.

2.1 Review: Dilations and Similarity   This section reviews the nonrigid transformation, the dilation, studied in previous mathematics courses, as well as proportional reasoning to examine the concept of similarity. We see similarity regularly in our everyday lives. For example, photos generally do not show an image that is true to size. Sometimes they show a much smaller version of the image and sometimes, especially in nature and science, photos show a much larger image of the actual object. The resources in this section provide explanatory content as well as opportunities to investigate the transformations and how they relate to similarity and opportunities to apply this knowledge in problem-solving situations.

2.1.1 Ratios and Proportions   - Explanation: CK-12: Geometry: “Ratios and Proportions” Link: CK-12: Geometry: “Ratios and Proportions” (HTML)

Instructions: Please click on the link above, and read and complete the example exercises as they arise. Solutions are explained following each example. While reading, consider these questions: What is a ratio? What is a proportion? Where do you see ratios and proportions in your everyday life?

Reading the material and completing the example problems should take approximately 30 minutes.

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[CCSS.Math.Content.7.RP.A.2](http://www.corestandards.org/Math/Content/7/RP/A/2)
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[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)
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[CCSS.ELA-Literacy.WHST.9-10.1e](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/e)

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• Did I Get This? Activity: Khan Academy’s “Proportions 1” and “Proportions 2” Link: Khan Academy’s “Proportions 1” and “Proportions 2” (HTML)

Instructions: Please click on the links above. Each one provides a series of practice problems that allow you to practice solving proportions. Each question has explanations for the example, worked out step-by-step if you need hints along the way. The program monitors your success, and when you have correctly completed six questions consecutively, it will tell you that you are ready to move on to a new skill.

Completing this activity should take approximately 30 minutes.

●      CCSS.Math.Content.7.RP.A.2

2.1.2 Definition of Similar   - Explanation: CK-12: Geometry: “Similar Polygons” Link: CK-12: Geometry: “Similar Polygons” (HTML)

Instructions: Please click on the link above, and scroll down to the sections titled “Similar Polygons” and “Scale Factors.” Read the content and complete the example problems as you go. Solutions are explained following each example. While reading, consider these questions: What makes two figures similar? What information must you know in order to determine their similarity? How do proportions relate to similarity?

Reading the material and completing the example problems should take approximately 30 minutes.

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[CCSS.Math.Content.HSG-SRT.A.2](http://www.corestandards.org/Math/Content/HSG/SRT/A/2)
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[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)
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[CCSS.ELA-Literacy.WHST.9-10.1e](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/e)

displayed on the webpage above.
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2.1.3 Similarity Transformations   - Explanation: CK-12: Geometry: “Similarity Transformations” Link: CK-12: Geometry: “Similarity Transformations” (HTML)

Instructions: Please click on the link above, and scroll down to the section titled “Dilations.” Read the content and complete the example problems as you go. Solutions are explained following each example. Upon completion of this reading, you should be able to determine if a transformation is or is not an example of dilation. Additionally, you should be able to draw dilations around the origin, given the scale factor.

Reading the material and completing the example problems should take approximately 15 minutes.

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[CCSS.Math.Content.HSG-SRT.A.1](http://www.corestandards.org/Math/Content/HSG/SRT/A/1)
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[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)
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[CCSS.ELA-Literacy.WHST.9-10.1e](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/e)

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• Did I Get This? Activity: Wikispaces: Justin Allen’s “U3—Lesson 3 hw—Dilations” Link: Wikispaces: Justin Allen’s “U3—Lesson 3 hw—Dilations” (DOCX)

Completing this activity should take approximately 30 minutes.

●      CCSS.Math.Content.HSG-SRT.A.1

2.2 Similarity Proofs   In this section, proofs continue to be practiced and used to further investigate geometric properties and concepts. Here proofs for similarity of triangles are explained. The resources in this section include explanatory content with embedded examples with solutions.

2.2.1 Similarity by AA   - Explanation: CK-12: Geometry: “Similarity by AA” Link: CK-12: Geometry: “Similarity by AA” (HTML)

Instructions: Please click on the link above, and scroll down to read the sections “Angles in Similar Triangles” and “Indirect Measure.” Complete the example problems as you go. The solutions are explained after each problem. Upon completion of the reading, explain in your own words why knowing the measures of two pairs of corresponding angles is sufficient information to determine similarity.

Reading this section and taking notes should take approximately 25 minutes.

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[CCSS.Math.Content.HSG-SRT.A.3](http://www.corestandards.org/Math/Content/HSG/SRT/A/3)
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[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)
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[CCSS.ELA-Literacy.WHST.9-10.1d](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/d)
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[CCSS.ELA-Literacy.WHST.9-10.1e](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/e)

displayed on the webpage above.
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2.2.2 Similarity by SSS and SAS   - Explanation: CK-12: Geometry: “Similarity by SSS and SAS” Link: CK-12: Geometry: “Similarity by SSS and SAS” (HTML)

Instructions: Please click on the link above, and scroll down to read the sections “SSS for Similar Triangles,” “SAS for Similar Triangles,” and “Similar Triangles Summary.” Complete the example problems as you go. The solutions are explained after each problem. Upon completion of the reading, explain in your own words why knowing the measures of three pairs of corresponding sides is sufficient information to determine similarity. Additionally, explain in your own words why knowing the measures of two pairs of corresponding sides with a pair of corresponding angles is sufficient information to determine similarity.

Reading this section and taking notes should take approximately 25 minutes.

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[CCSS.Math.Content.HSG-SRT.A.3](http://www.corestandards.org/Math/Content/HSG/SRT/A/3)
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[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)
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[CCSS.ELA-Literacy.WHST.9-10.1d](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/d)
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[CCSS.ELA-Literacy.WHST.9-10.1e](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/e)

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2.3 Trigonometry   In this subunit, we revisit the Pythagorean theorem. With this introduction, the subunit segues into special right triangles, similar right triangles, and finally, trigonometry. Finally, the subunit ends with an explanation of the laws of sines and cosines, combining the understanding of Pythagorean theorem with trigonometry to find unknown measures of right triangles. The resources of this section include explanatory content and practice exercises that provide immediate feedback.

2.3.1 The Pythagorean Theorem   - Explanation: CK-12: Geometry: “The Pythagorean Theorem” and “Converse of the Pythagorean Theorem” Link: CK-12: Geometry: “The Pythagorean Theorem” and “Converse of the Pythagorean Theorem” (HTML)

Instructions: Please click on each link above, and read the content provided. At “The Pythagorean Theorem” link, please read the sections titled “The Pythagorean Theorem,” “Another Proof of the Pythagorean Theorem,” “Using the Pythagorean Theorem,” and “Pythagorean Triples.” At the “Converse of the Pythagorean Theorem” link, please read the sections titled “Converse of the Pythagorean Theorem” and “Identifying Acute and Obtuse Triangles.” Complete the example problems as you go. The solutions are explained following each problem. Upon completion of the reading, summarize the Pythagorean theorem and provide a visual to explain how this theorem works.

Reading this section and taking notes should take approximately 20 minutes.

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[CCSS.Math.Content.HSG-SRT.B.4](http://www.corestandards.org/Math/Content/HSG/SRT/B/4)
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[CCSS.Math.Content.HSG-SRT.C.8](http://www.corestandards.org/Math/Content/HSG/SRT/C/8)
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[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)
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[CCSS.ELA-Literacy.WHST.9-10.1e](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/e)

displayed on the webpage above.
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Instructions: Please click on the link above. It provides a series of practice problems that allow you to use the Pythagorean theorem to solve for an unknown side. Each question has explanations for the example, worked out step-by-step if you need hints along the way. The program monitors your success, and when you have correctly completed six questions consecutively, it will tell you that you are ready to move on to a new skill.

Completing this activity should take approximately 15 minutes.

●      CCSS.Math.Content.HSG-SRT.B.4
●      CCSS.Math.Content.HSG-SRT.C.8

2.3.2 Similar Right Triangles   - Explanation: CK-12: Geometry: “Using Similar Right Triangles” Link: CK-12: Geometry: “Using Similar Right Triangles” (HTML)

Instructions: Please click on the link above, and read the content in the sections titled “Inscribed Similar Triangles” and “Geometric Mean.” Complete the example problems as you go. The solutions are explained following each problem. Upon completion of this reading, summarize the reading and be sure you can solve for unknown side lengths of triangles, using geometric mean.

Reading this section and taking notes should take approximately 30 minutes.

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[CCSS.Math.Content.HSG-SRT.C.8](http://www.corestandards.org/Math/Content/HSG/SRT/C/8)
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[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)
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[CCSS.ELA-Literacy.WHST.9-10.1e](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/e)

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2.3.3 Special Right Triangles   - Explanation: CK-12: Geometry: “Special Right Triangles” Link: CK-12: Geometry“Special Right Triangles” (HTML)

Instructions: Please click on the link above, and read the content in the sections titled “Isosceles Right Triangles” and “30-60-90 Triangles.” Complete the example problems as you go. The solutions are explained following each problem. Upon completion of the reading, you should be able to identify the two special right triangles introduced in the reading, identify the relationship that exists among the sides of each triangle, and use these ratios to find the length of an unknown side.

Reading this section and taking notes should take approximately 20 minutes.

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[CCSS.Math.Content.HSG-SRT.C.6](http://www.corestandards.org/Math/Content/HSG/SRT/C/6)
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[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)
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[CCSS.ELA-Literacy.WHST.9-10.1e](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/e)

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• Did I Get This? Activity: Khan Academy’s “Special Right Triangles” Link: Khan Academy’s “Special Right Triangles” (HTML)

Instructions: Please click on the link above. It provides a series of practice problems that allow you to use the Pythagorean theorem to solve for an unknown side. Each question has explanations for the example, worked out step-by-step if you need hints along the way. The program monitors your success, and when you have correctly completed six questions consecutively, it will tell you that you are ready to move on to a new skill.

Completing this activity should take approximately 15 minutes.

●      CCSS.Math.Content.HSG-SRT.C.6

2.3.4 Tangent, Sine, and Cosine   - Explanation: CK-12: Geometry: “Tangent, Sine and Cosine” Link: CK-12: Geometry“Tangent, Sine and Cosine” (HTML)

Instructions: Please click on the link above, and read the content in the sections entitled “What is Trigonometry?,” “Sine, Cosine, and Tangent Ratios,” “Finding the Sides of a Triangle Using Trig Ratios,” and “Angles of Depression and Elevation.” Complete the example problems as you go. The solutions are explained following each problem.

Watching this lecture and taking notes should take approximately 30 minutes.

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[CCSS.Math.Content.HSG-SRT.C.6](http://www.corestandards.org/Math/Content/HSG/SRT/C/6)
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[CCSS.Math.Content.HSG-SRT.C.8](http://www.corestandards.org/Math/Content/HSG/SRT/C/8)
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[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)
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[CCSS.ELA-Literacy.WHST.9-10.1e](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/e)

displayed on the webpage above.
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Instructions: Please click on the link above. It provides a series of practice problems that ask you to identify the trigonometric ratios. Each question has explanations for the example, worked out step-by-step if you need hints along the way. The program monitors your success, and when you have correctly completed six questions consecutively, it will tell you that you are ready to move on to a new skill.

Completing this activity should take approximately 15 minutes.

●      CCSS.Math.Content.HSG-SRT.C.6
●      CCSS.Math.Content.HSG-SRT.C.8

Instructions: Please click on the link above. It provides a series of practice problems that require you to find the length of unknown sides given the lengths of other sides and the trigonometric ratios. Each question has explanations for the example, worked out step-by-step if you need hints along the way. The program monitors your success, and when you have correctly completed six questions consecutively, it will tell you that you are ready to move on to a new skill.

Completing this activity should take approximately 15 minutes.

●      CCSS.Math.Content.HSG-SRT.C.6
●      CCSS.Math.Content.HSG-SRT.C.8

2.3.5 Inverse Trigonometric Ratios   - Explanation: CK-12: Geometry: “Inverse Trigonometric Ratios” Link: CK-12: Geometry: “Inverse Trigonometric Ratios” (HTML)

Instructions: Please click on the link above, and read the content in the sections titled “Inverse Trigonometric Ratios,” “Solving Triangles,” and “Real-Life Situations.” Complete the example problems as you go. The solutions are explained following each problem.

Watching this lecture and taking notes should take approximately 30 minutes.

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[CCSS.Math.Content.HSG-SRT.C.6](http://www.corestandards.org/Math/Content/HSG/SRT/C/6)
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[CCSS.Math.Content.HSG-SRT.C.8](http://www.corestandards.org/Math/Content/HSG/SRT/C/8)
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[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)
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[CCSS.ELA-Literacy.WHST.9-10.1e](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/e)

displayed on the webpage above.
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2.3.6 Laws of Sines and Cosines   - Checkpoint: Wikispaces: precalculusnwr7’s “Solve Bearing and Orienteering Problems with Law of Sines and Law of Cosines” Link: Wikispaces: precalculusnwr7’s “Solve Bearing and Orienteering Problems with Law of Sines and Law of Cosines” (HTML)

Instructions: Please click on the link above. This page will give you an opportunity to apply your knowledge in a new and challenging context. The concepts of the laws of sine and cosine are reviewed for you, and the concept of bearing and how it relates to these mathematical concepts are explained. Read carefully and take notes. At the bottom of the page below the subtitle, “Applying what we know to solve bearing and orienteering problems with Law of Sines and Law of Cosines,” are two application problems. Attempt to solve them on your own first. If you are struggling with the first one, read through the explanation or play the video explanation provided at the bottom of the page. Using this explanation, attempt the second problem on your own and then compare your work to the explanation provided.

Completing this activity should take approximately 30 minutes.

●      CCSS.Math.Content.HSG-SRT.D.10
●      CCSS.Math.Content.HSG-SRT.D.11
●      CCSS.ELA-Literacy.WHST.9-10.1d
●      CCSS.ELA-Literacy.WHST.9-10.1e
●      CCSS.ELA-Literacy.RST.9-10.3
●      CCSS.ELA-Literacy.RST.9-10.4
●      CCSS.ELA-Literacy.RST.9-10.5

• Explanation: CK-12: Geometry: “Extension: Laws of Sines and Cosines” Link: CK-12: Geometry: “Extension: Laws of Sines and Cosines” (HTML)

Instructions: Please click on the link above, and read the content in the sections titled “Law of Sines” and “Law of Cosines.” Complete the example problems as you go. The solutions are explained following each problem. As you read, consider these questions: What information is required in order to apply the law of sines or the law of cosines? What theorem is used to determine these laws?

Watching this lecture and taking notes should take approximately 30 minutes.

●      CCSS.Math.Content.HSG-SRT.D.10
●      CCSS.Math.Content.HSG-SRT.D.11
●      CCSS.ELA-Literacy.RST.9-10.3
●      CCSS.ELA-Literacy.WHST.9-10.1e