Unit 2

Similarity and Trigonometry

 

Geometry

 

Unit Description:

 

Students apply their earlier experience with dilations and proportional reasoning to build a formal understanding of similarity. They identify criteria for similarity of triangles, use similarity to solve problems, and apply similarity in right triangles to understand right triangle trigonometry, with particular attention to special right triangles and the Pythagorean Theorem.

 

Standards for Mathematical Practice      

MP.1 Make sense of problems and persevere in solving them.

MP.2 Reason abstractly and quantitatively.

MP.3 Construct viable arguments and critique the reasoning of others.

MP.4 Model with mathematics.

MP.5 Use appropriate tools strategically.

MP.6 Attend to precision.

MP.7 Look for and make use of structure.

Louisiana Student Standards for Mathematics (LSSM)

G-SRT: Similarity, Right Triangles, and Trigonometry

A. Understand similarity in terms of similarity transformations

G-SRT.A.1

 

 

Verify experimentally the properties of dilations given by a center and a scale factor. Include centers not at the origin.

a. 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.

b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

*I can describe the undefined terms: point, line, and distance along a line in a plane.

*I can define circle and the distance around a circular arc.

G-SRT.A.2

 

 

Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; 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.

*I can describe the different types of transformations including translations, reflections, rotations and dilations.

*I can describe transformations as functions that take points in the coordinate plane as inputs and give other points as outputs.

*I can compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).

*I can represent transformations in the plane using, e.g., transparencies and geometry software.

*I can write functions to represent transformations.

G-SRT.A.3

 

 

Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

*I can describe the rotations and/or reflections that carry it onto itself given a rectangle, parallelogram, trapezoid, or regular polygon.

 

B. Prove and apply theorems involving similarity.

G-SRT.B.4

 

 

Prove and apply theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity; SAS similarity criteria, SSS similarity criteria, AA similarity criteria.

*I can recall definitions of angles, circles, perpendicular and parallel lines and line segments.

*I can develop definitions of rotations, reflections and translations in terms of angles, circles, perpendicular lines, parallel lines and line segments.

G-SRT.B.5

 

 

Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

*I can, given a geometric figure and a rotation, reflections or translation, draw the transformed figure using, e.g. graph paper, tracing paper or geometry software.

*I can a draw transformed figure and specify the sequence of transformations that were used to carry the given figure onto the other.

C. Define trigonometric ratios and solve problems involving right triangles.

G-SRT.C.6

 

 

Understand that by similarity, side ratios in right triangles, including special right triangles (30-60-90 and 45-45-90), are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

*I can use geometric descriptions of rigid motions to transform figures.

*I can predict the effect of a given rigid motion on a given figure.

*I can define congruence in terms of rigid motions (i.e. two figures are congruent if there exists a rigid motion, or composition of rigid motions, that can take one figure to the second).

*I can describe rigid motion transformations.

*I can predict the effect of a given rigid motion.

*I can decide if two figures are congruent in terms of rigid motions (it is not necessary to find the precise transformation(s) that took one figure to a second, only to understand that such a transformation or composition exists).

*I can, given two figures, use the definition of congruence in terms of rigid motion to decide if they are congruent.

G-SRT.C.7

 

 

Explain and use the relationship between the sine and cosine of complementary angles.

*I can identify corresponding angles and sides of two triangles.

*I can identify corresponding pairs of angles and sides of congruent triangles after rigid motions.

*I can use the definition of congruence in terms of rigid motions to show that two triangles are congruent if corresponding pairs of sides and corresponding pairs of angles are congruent.

*I can use the definition of congruence in terms of rigid motions to show that if the corresponding pairs of sides and corresponding pairs of angles of two triangles are congruent then the two triangles are congruent.

*I can justify congruency of two triangles using transformations.

G-SRT.C.8

 

 

Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

*I can formally use dynamic geometry software or straightedge and compass to take angles to angles and segments to segments.

*I can identify ASA, SAS, and SSS.

*I can explain how the criteria for triangle congruence (ASA, SAS, SSS) follows from the definition of congruence in terms of rigid motions (i.e. if two angles and the included side of one triangle are transformed by the same rigid motion(s) then the triangle image will be congruent to the original triangle).

Additional Standards for Honors Classes

G-SRT.D.9 (+) Derive the formula  for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.

 

G-SRT.D.10 (+) Prove the Laws of Sines and Cosines and use them to solve problems.

 

G-SRT.D.11 (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).

G-MG: Modeling with Geometry

A. Apply geometric concepts in modeling situations.

G-MG.A.1

 

 

Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).

*I can identify the hypothesis and conclusion of a triangle sum theorem.

*I can identify the hypothesis and conclusion of a base angle of isosceles triangles.

*I can identify the hypothesis and conclusion of midsegment theorem.

*I can identify the hypothesis and conclusion of points of concurrency.

*I can design an argument to prove theorems about triangles.

*I can analyze components of the theorem.

*I can prove theorems about triangles

G-MG.A.2

 

 

Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).

*I can classify types of quadrilaterals.

*I can explain theorems for various parallelograms involving opposite sides and angles and relate to figure.

*I can explain theorems for various parallelograms involving diagonals and relate to figure.

*I can use the principle that corresponding parts of congruent triangles are congruent to solve problems.

*I can use properties of special quadrilaterals in a proof.

G-MG.A.3

 

 

Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).

G-CO: Congruence

A. Experiment with transformations in the plane.

G-CO.A.2

 

 

Represent transformations in the plane using, e.g. transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).

-Within this unit add the concept of similarity versus congruence as set forth in Unit 1.

*I can describe the different types of transformations including translations, reflections, rotations and dilations.

D. Make geometric constructions.

G-CO.D.12

 

 

Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

*I can use the definition of dilation to construct a similar figure.

 

 

 

Enduring Understandings:

*Transformations, symmetry, and spatial reasoning can be used to analyze and model mathematical situations.

*Proportional relationships express how quantities change in relationship to each other.

*Characteristics, properties, and mathematical arguments about geometric relationships can be analyzed and developed using logical and spatial reasoning.

*Similarity can be demonstrated using logical reasoning.

*Similarity in polygons has real-life application in a variety of areas including art, architecture, and sciences.

*Different observed relationships between geometric objects are provable using basic geometric building blocks and previously proven relationships between these building blocks and between other geometric objects.

*Triangles are fundamental aesthetic, structural elements that are useful in many disciplines such as art, architecture, and engineering.

*Developing techniques for measuring indirectly is a useful in many aspects of daily life.

*Proportional relationships express how quantities change in relationship to each other.

*Trigonometry can help us solve real world problems that involve triangles.

Essential Questions:

*How does geometry explain or describe the structure of our world?

*How does my understanding of algebraic principles help me solve geometric problems?

*How are the concepts of similarity and congruence related to each other?

*What special relationships occur between congruent triangles and similar triangles?

*How can drawings and figures be used to justify arguments and conjectures about congruence and similarity?

*How can proportions be used to solve problem involving similarity?

*What special properties exist in a right triangle that makes it unique?

*How can special segments of a triangle be used to solve real-world problems?

*What connections can be made between algebraic concepts and geometric concepts?

*How are the trigonometric ratios useful in modeling real life situations?