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.[[2]](#_ftn1)
[2]Ibid
Unit 2 Time Advisory
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.
Standards Addressed (Common Core): - CCSS.Math.Content.7.RP.A.2 - CCSS.Math.Content.HSG-SRT.A.1 - CCSS.Math.Content.HSG-SRT.A.2 - CCSS.Math.Content.HSG-SRT.A.3 - CCSS.Math.Content.HSG-SRT.B.4 - CCSS.Math.Content.HSG-SRT.B.5 - CCSS.Math.Content.HSG-SRT.C.6 - CCSS.Math.Content.HSG-SRT.C.7 - CCSS.Math.Content.HSG-SRT.C.8 - CCSS.Math.Content.HSG-MG.A.1 - CCSS.Math.Content.HSG-MG.A.2 - CCSS.Math.Content.HSG-MG.A.3 - CCSS.Math.Content.HSG-SRT.D.9 - CCSS.Math.Content.HSG-SRT.D.10 - CCSS.Math.Content.HSG-SRT.D.11 - CCSS.Math.Content.HSG-MG.A.1 - CCSS.Math.Content.HSG-MG.A.2 - CCSS.Math.Content.HSG-MG.A.3 - 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
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.
Standards Addressed (Common Core):
●
[CCSS.Math.Content.7.RP.A.2](http://www.corestandards.org/Math/Content/7/RP/A/2)
●
[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)
●
[CCSS.ELA-Literacy.WHST.9-10.1e](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/e)
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
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.
Standards Addressed (Common Core):● CCSS.Math.Content.7.RP.A.2
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
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.
Standards Addressed (Common Core):
●
[CCSS.Math.Content.HSG-SRT.A.2](http://www.corestandards.org/Math/Content/HSG/SRT/A/2)
●
[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)
●
[CCSS.ELA-Literacy.WHST.9-10.1e](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/e)
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
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.
Standards Addressed (Common Core):
●
[CCSS.Math.Content.HSG-SRT.A.1](http://www.corestandards.org/Math/Content/HSG/SRT/A/1)
●
[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)
●
[CCSS.ELA-Literacy.WHST.9-10.1e](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/e)
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
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)
Instructions: Please click on the link above, download the lesson entitled “U3—Lesson 3 hw—Dilations,” and complete the problems. When you finish, download the answers to the activity to check your work.
Completing this activity should take approximately 30 minutes.
Standards Addressed (Common Core):● CCSS.Math.Content.HSG-SRT.A.1
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
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.
Standards Addressed (Common Core):
●
[CCSS.Math.Content.HSG-SRT.A.3](http://www.corestandards.org/Math/Content/HSG/SRT/A/3)
●
[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)
●
[CCSS.ELA-Literacy.WHST.9-10.1d](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/d)
●
[CCSS.ELA-Literacy.WHST.9-10.1e](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/e)
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
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.
Standards Addressed (Common Core):
●
[CCSS.Math.Content.HSG-SRT.A.3](http://www.corestandards.org/Math/Content/HSG/SRT/A/3)
●
[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)
●
[CCSS.ELA-Literacy.WHST.9-10.1d](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/d)
●
[CCSS.ELA-Literacy.WHST.9-10.1e](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/e)
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
- Checkpoint: Mathematics Assessment Project: “Solving Geometry
Problems: Floodlights”
Link: Mathematics Assessment Project: “Solving Geometry Problems:
Floodlights” (HTML)
Instructions: Please click on the link above, and scroll down to the section of the lesson entitled “Resources.” Download the PDF file underneath the heading “Lesson (complete).” The assessment is on page S-1 and is titled “Floodlights.” Complete the questions to the best of your ability. When you are finished, use the solutions on page T-10 to check your understanding. For an additional challenge, scroll down to the sample responses. Questions are provided to guide you in the analysis of the work and in making improvements to each sample.
Completing this activity should take approximately 30 minutes. The extension will add another 30 minutes.
Standards Addressed (Common Core):
● CCSS.Math.Content.HSG-CO.C.9
● CCSS.Math.Content.HSG-CO.C.10
● CCSS.Math.Content.HSG-SRT.A.1
● CCSS.Math.Content.HSG-SRT.A.2
● CCSS.Math.Content.HSG-SRT.A.3
● CCSS.Math.Content.7.RP.A.2
● CCSS.ELA-Literacy.RST.9-10.3
● CCSS.ELA-Literacy.RST.9-10.4
● CCSS.ELA-Literacy.RST.9-10.5
● CCSS.ELA-Literacy.WHST.9-10.1d
● CCSS.ELA-Literacy.WHST.9-10.1e
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
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.
Standards Addressed (Common Core):
●
[CCSS.Math.Content.HSG-SRT.B.4](http://www.corestandards.org/Math/Content/HSG/SRT/B/4)
●
[CCSS.Math.Content.HSG-SRT.C.8](http://www.corestandards.org/Math/Content/HSG/SRT/C/8)
●
[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)
●
[CCSS.ELA-Literacy.WHST.9-10.1e](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/e)
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
Did I Get This? Activity: Khan Academy’s “Pythagorean Theorem” Link: Khan Academy’s “Pythagorean Theorem” (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.
Standards Addressed (Common Core):● CCSS.Math.Content.HSG-SRT.B.4
● CCSS.Math.Content.HSG-SRT.C.8
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
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.
Standards Addressed (Common Core):
●
[CCSS.Math.Content.HSG-SRT.C.8](http://www.corestandards.org/Math/Content/HSG/SRT/C/8)
●
[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)
●
[CCSS.ELA-Literacy.WHST.9-10.1e](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/e)
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
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.
Standards Addressed (Common Core):
●
[CCSS.Math.Content.HSG-SRT.C.6](http://www.corestandards.org/Math/Content/HSG/SRT/C/6)
●
[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)
●
[CCSS.ELA-Literacy.WHST.9-10.1e](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/e)
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
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.
Standards Addressed (Common Core):● CCSS.Math.Content.HSG-SRT.C.6
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
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.
Standards Addressed (Common Core):
●
[CCSS.Math.Content.HSG-SRT.C.6](http://www.corestandards.org/Math/Content/HSG/SRT/C/6)
●
[CCSS.Math.Content.HSG-SRT.C.8](http://www.corestandards.org/Math/Content/HSG/SRT/C/8)
●
[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)
●
[CCSS.ELA-Literacy.WHST.9-10.1e](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/e)
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
Did I Get This? Activity: Khan Academy’s “Trigonometry 1” Link: Khan Academy’s “Trigonometry 1” (HTML)
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.
Standards Addressed (Common Core):● CCSS.Math.Content.HSG-SRT.C.6
● CCSS.Math.Content.HSG-SRT.C.8
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.Did I Get This? Activity: Khan Academy’s “Trigonometry 2” Link: Khan Academy’s “Trigonometry 2” (HTML)
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.
Standards Addressed (Common Core):● CCSS.Math.Content.HSG-SRT.C.6
● CCSS.Math.Content.HSG-SRT.C.8
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
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.
Standards Addressed (Common Core):
●
[CCSS.Math.Content.HSG-SRT.C.6](http://www.corestandards.org/Math/Content/HSG/SRT/C/6)
●
[CCSS.Math.Content.HSG-SRT.C.8](http://www.corestandards.org/Math/Content/HSG/SRT/C/8)
●
[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)
●
[CCSS.ELA-Literacy.WHST.9-10.1e](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/e)
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
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.
Standards Addressed (Common Core):
●
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
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
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.
Standards Addressed (Common Core):● 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
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.Did I Get This? Activity: Khan Academy’s “Law of Cosines” Link: Khan Academy’s “Law of Cosines” (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 30 minutes.
Standards Addressed (Common Core):● CCSS.Math.Content.HSG-SRT.D.10
● CCSS.Math.Content.HSG-SRT.D.11
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.