GSRT:
Similarity, Right Triangles, and Trigonometry

A. Understand
similarity in terms of similarity transformations

GSRT.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.

GSRT.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.

GSRT.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.

GSRT.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.

GSRT.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.

GSRT.C.6

Understand that by similarity, side ratios in right
triangles, including special right triangles (306090 and 454590), 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.

GSRT.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.

GSRT.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

GSRT.D.9 (+) Derive the formula for the area of a triangle by drawing an
auxiliary line from a vertex perpendicular to the opposite side.
GSRT.D.10 (+) Prove the Laws of Sines and Cosines and
use them to solve problems.
GSRT.D.11 (+) Understand and apply the Law of Sines and
the Law of Cosines to find unknown measurements in right and nonright
triangles (e.g.,
surveying problems, resultant forces).

GMG:
Modeling with Geometry

A. Apply geometric concepts in modeling
situations.

GMG.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

GMG.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.

GMG.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).^{ ★}

GCO:
Congruence

A.
Experiment with transformations in the plane.

GCO.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.

GCO.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.
