Download Standard 4.2 Similarity

Curriculum Guide
High School – Math Standard 4.2
Math.4.2 - Concepts of similarity are foundational to geometry and its applications.
Related Colorado Department of Education Sample Units:







Algebra II - Trickster Trigonometry (Concepts: unit circle, coordinate plane, trigonometric functions, angles, model, periodic phenomena)
Geometry - Identical Twins And Mini-Me (Concepts: dilation, center, transformation, scale factor, magnitude, direction, congruence,
corresponding angles and sides, proportionality, rigid transformation)
Geometry - 3 Rights Don’t Make A…(Concepts: sides ratios, angles, right triangle, trigonometric functions, similar triangles)
Geometry - What Goes Around (Concepts: arc length, inscribed angles, circumscribed angles, central angles, circles, center, radius,
equation, chords, arcs, proportionally)
Integrated Math 2 - Duck, Duck, Goose (Concepts: ratio, corresponding sides, corresponding angles, scale factor, similar figures, dilations,
center of dilation, trigonometric ratios, right triangles, trigonometric functions)
Integrated Math 3 - Within And Around (Concepts: arc length, inscribed angles, circumscribed angles, central angles, circles, center, radius,
equation, chords, arcs, proportionally, proofs, geometric constructions, conjecture, coordinate plane, geometric relationships)
Integrated Math 3 - Sine Of The Times (Concepts: unit circle, coordinate plane, trigonometric functions, angles, model, periodic phenomena)
Essential Questions - 21st Century Skills and Readiness Competencies (District):
1a. What are the key properties of a dilation?
1a. In general, how does a dilation transform a figure?
1a. How do dilations affect lines and line segments?
1a. Do you think dilations are rigid motion? Why or why not?
1a. Can you experimentally verify the properties of dilations given by the center and a
scale factor?
1ai. Can you show that a dilation takes a line not passing through the center of the
dilation to a parallel line, and
leaves a line passing through the center unchanged?
1aii. Can you show that the dilation of a line segment is longer or shorter in the ratio
given by the scale factor?
1b. What is the difference between congruence and similarity?
1b. How do you know if two or more figures are similar?
1c. How do you determine if two triangles are similar to one another (i.e., What is the
relationship between the corresponding sides and angles of similar triangles?)
1d. How can you use similarity transformations to prove the AA criteria for triangle
similarities?
2a. Can you prove a line parallel to one side of a triangle divides the other two
proportionally?
2a. Can you prove that the medians of a triangle meet at a point?
2a. Can you use triangle similarity to prove the Pythagorean Theorem?
2b. Are all circles similar? If so, how can you prove this?
Evidence Outcomes (District):
1. Understand similarity in terms of similarity transformations.
a. Verify experimentally the properties of dilations given by a center and a
scale factor.
i. Show that a dilation takes a line not passing through the center of the
dilation to a parallel line, and leaves a line passing through the center
unchanged.
ii. Show that the dilation of a line segment is longer or shorter in the
ratio given by the scale factor.
b. Given two figures, use the definition of similarity in terms of similarity
transformations to decide if they are similar.
c. Explain using similarity transformations the meaning of similarity for triangles
as the equality of all corresponding pairs of angles and the proportionality of all
corresponding pairs of sides.
d. Use the properties of similarity transformations to establish the AA criterion
for two triangles to be similar.
2. Prove theorems involving similarity.
a. Prove theorems about triangles.
b. Prove that all circles are similar.
c. Use congruence and similarity criteria for triangles to solve problems and to
prove relationships in geometric figures.
3. Define trigonometric ratios and solve problems involving right triangles.
a. Explain that by similarity, side ratios in right triangles are properties of the
2c. How can you use similarity and congruence criteria to solve real-world
applications?
3a. How does similarity give rise to trigonometric ratios?
3b. How do the trigonometric ratios of complementary angles relate to one another?
3c. How can the Pythagorean Theorem be used to solve problems involving triangles?
4a. What is a trigonometric identity? Why are some identities called trigonometric
identities?
4a. How is the radian measure of an angle related to the length of the arc on the unit
circle?
4a. How are the values of trigonometric functions derived using the unit circle?
4b. When are trigonometric identities used to determine angle measures?
4b. What are the similarities and differences between the values of the trigonometric
functions in all four quadrants?
4b. Why is the unit circle important in the study of trigonometry?
4b. What is the relationship between the unit circle and the Cartesian Coordinates
System?
5a. What is a central angle? What is the relationship between a central angle and its
intercepted arc?
5a. What is an inscribed angle? What is the relationship between an inscribed angle
and its intercepted arc?
5a. What type of angle results in an inscribed angle whose endpoints are those of the
diameter of a circle?
5a. What is the relationship between a radius (or diameter) and a tangent of a
circle? What is the point where they meet named?
5a. What is the relationship between congruent chords and their intercepted arcs?
5a. If two chords in a circle are congruent, what is their
relationship regarding distance from the center of the circle?
5a. If a radius is perpendicular to a chord, what else must be true?
5a. What is the relationship between two chords that intersect in a circle?
5b. What does it mean for a circle to be inscribed in a triangle? How would
you construct this?
5b. What does it mean for a circle to be circumscribed about a triangle? How
would you construct this?
5b. What are points of concurrency? How does this relate to constructing a circle
inscribed in a triangle?
5b. What are incenter, circumcenter and centroid? How do these terms relate to circles
inscribed in or circumscribed about a triangle?
5c. What do you know about the angles of a quadrilateral inscribed in a circle? How
can you prove this to be true?
6a. How is arc length related to the radius and central angle of a circle?
6a. How is arc length related to circumference of a circle?
6a. How can you calculate arc length?
6b. How is the area of a sector related to the radius and central angle of a circle?
6b. How is area of a sector related to area of a circle?
6b. How can you calculate the area of a sector?
angles in the triangle, leading to definitions of trigonometric ratios for acute
angles.
b. Explain and use the relationship between the sine and cosine of
complementary angles.
c. Use trigonometric ratios and the Pythagorean Theorem to solve right
triangles in applied problems.
4. Prove and apply trigonometric identities.
a. Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1.
b. Use the Pythagorean identity to find sin(θ), cos(θ), or tan(θ) given sin(θ),
cos(θ), or tan(θ) and the quadrant of the angle.
5. Understand and apply theorems about circles.
a. Identify and describe relationships among inscribed angles, radii, and
chords.
b. Construct the inscribed and circumscribed circles of a triangle.
c. Prove properties of angles for a quadrilateral inscribed in a circle.
6. Find arc lengths and areas of sectors of circles.
a. Using similarity, derive the fact that the length of the arc intercepted by an
angle is proportional to the radius, and define the radian measure of the angle
as the constant of proportionality.
b. Derive the formula for the area of a sector.
Academic Vocabulary (District):
AA Similarity
arc length
central angle
centroid
chord
circumcenter
circumscribed
Assessment (District):
Informal Checks for Understanding: Accuracy and thoroughness of student notes,
study guides, and graphic organizers will demonstrate informal evidence of concept
processing.
Observation/Dialogue: Teacher observation and monitoring of the frequency and
quality of student contributions to discussion and the sophistication of student
responses to critical questioning will serve as informal evidence of concept processing
co-functions
complementary angles
constant of proportionality
cosecant
cosine
cotangent
dilation
equivalent forms of trig identities
incenter
inscribed angle
intercepted arc
point of tangency
Pythagorean Identity
Pythagorean Theorem
quadrilateral
radian measure
radians
radii
ratio
scale factor
secant
secant
sector
similarity
Simplifying/Rationalizing answers
sine
supplementary angles
tangent
trigonometric identities
trigonometric ratios
unit circle
and skill development.
Suggested Activities/Strategies (District):
1a. In Dilating a Line, students experimentally verify the properties of dilations given by
a center and a scale factor.

1a. Dilating a Line
1b. Given two circles, students find the translation rule and the scale factor of the
dilation.

1b. Similar Circles
1b, 1c. In this problem, students are given a picture of two triangles that appear to be
similar, but whose similarity cannot be proven without further information. Asking
students to provide a sequence of similarity transformations that maps one triangle to
the other requires them to apply the definition of similarity in terms of similarity
transformations.

1b, 1c. Are They Similar?
1c. This online lesson requires students to calculate missing lengths of similar
triangles.

1c. Finding Missing Sides of Similar Triangles
1d. In this lesson, students state the AA criterion for two triangles to be similar and
provide supporting examples with justifications to prove it that it holds true for all types
of triangles.

1d. AA Similarity Discovery Lesson
Resources/Technology (District):

1b-d. Triangle Similarity from Khan Academy

1d. AA Similarity Video

1d. Triangle Similarities: Proving AA
Quiz/Test: Formally measured evidence of outcomes and overall standard
achievement will be established via quizzes, unit tests, and benchmark tests.
Performance Tasks/Projects: Student self-evaluation and peer evaluation will be
utilized for informal feedback on at least one activity per standard. Performance tasks
and projects will be formally assessed by the teacher using rubrics for holistic
evaluation.





















1a. Dilations and Parallel Lines
1a. Dilations and Scale Factors
1b. Dilations Quiz
1c. Similar Triangles and Similarity Transformations
2b. Online Practice for Similar Circles
3, 4. All About Trigonometry
See: http://regentsprep.org/regents/math/geometry/math-GEOMETRY.htm
o
2a. Proofs with Similar Triangles
2a. Videos on Similar Triangles
2b. Similarity and Circles
3a. PowerPoint on Similar Right Triangles and Trig
3c. Trig Applications from Purple Math
4a, 4b. Trigonometric Identities
Similar Polygons
4a, 4b. Pythagorean Identity Proofs Videos
5a. Chords and Length Determination
5a. Inscribed and Circumscribed Circles
5a. Chords and Arcs
5b. YouTube Video on Inscribed and Circumscribed Circles of a Triangle
5c. Inscribed Quadrilateral Conjecture
5c. YouTube: Opposite Angles of a Quadrilateral Inscribed in a Circle
5c. Quadrilaterals Inscribed in Circles
2a. Students construct the centroid of a triangle, define median, use dynamic geometry
software to test their conjectures about the centroid of a triangle, and articulate the
properties of the centroid of the triangle.

2a. Lesson Investigating the Centroid of a Triangle
2a. Students prove the Pythagorean Theorem using similar triangles.

2a. Proving the Pythagorean Theorem Using Triangle Similarity
2b. Students prove that two circles are always similar.

2b. Practice on Similar Circles
2c. In the task High Scorer, students use similarity and congruence of triangles to
determine the placement of a ladder being used to repair a scoreboard in the school's
gymnasium.

2c. High Scorer Performance Task
2c. This task challenges students to use their knowledge of Pythagorean theorem to
show that given rhombuses are similar and knowledge of proportional sides or angle
size to justify that two different rhombuses are similar. Students construct a convincing
argument to show that given four‐sided figures are rhombuses.

2c. Rhombuses
2c, 3a-c. This task challenges students to use their understanding of similar triangles to
identify similar triangles on a grid and from dimensions. Students use trig ratios to
calculate an angle in a 3,4,5 right triangle and apply Pythagorean theorem to find
missing dimensions in right triangles. Students construct arguments to prove that
two triangles are similar.

2c, 3a-c. Hopewell Geometry
3b. These practice exercises emphasize that the two acute angles of right triangles are
complementary as students explore the relationships between the sine and cosine
values of the complementary angles.

3b. Complementary Angles and Trigonometric Ratios
3c. Right Triangle Applications provides practice in solving the Pythagorean Theorem,
calculating missing side lengths and angle measures using trigonometry, as well as
trigonometric word problems.

3c. Right Triangle Applications
3c. In this sports-based activity link, students learn properties of right triangles and
trigonometry and apply the properties in designing various sports complexes.

3c. Sports-Based Right Triangle Applications
4a, 4b. Students review and practice proving famous Pythagorean Identities.

4a, 4b. Famous IDs: Pythagoras Identities
4b. Students use their knowledge of trigonometric ratios to complete the puzzle.

4b. Trigonometry Puzzle
5a. In Broken Pottery, students develop a method to determine the diameter of a
circular plate when given a shard of that plate (a pottery remnant). Key concepts
include segment relationships in circles.

5a. Broken Pottery Activity
5b. This is a three-lesson unit that discovers and applies points of concurrency of a
triangle. The lessons are labs used to introduce the topics of incenter, circumcenter,
centroid, circumscribed circles, and inscribed circles.

5b. Searching for the Center
5c. In this lesson, students will explore the fact that opposite angles of a quadrilateral
inscribed in a circle must be supplementary.

5c. Inscribed Quadrilaterals
6a, 6b. Students use geometry concepts and theorems to calculate the area of a
shaded region of a circle.

See: MAA.org
o
6a, 6b. Arc Lengths and Areas of Sectors







5. Central Angles and Inscribed Angles Video
6a. Finding Arc Lengths in Radians
6b. Areas of Circles and Sectors
See Alabama Learning Exchange
Constructions Using Geometer's Sketch Pad
Geometry Resources by Topic
Shadow Problem
6a, 6b. Students work in cooperative groups to discover the relationships between arc
length, central angle measure, and circumference. They discover the relationship
between circle area, central angle measures, and sector area. Students share their
discoveries and create formulas for calculating arc length and sector area. Then they
practice these skills in a journal entry and by solving a real-life extension.
 See Alabama Learning Exchange





6a, 6b. ALEX Lesson Plan: Arc Lengths and Sector Areas