GCO:
Congruence

A.
Experiment with transformations in the plane.

GCO.A.1

Know precise
definitions of angle, circle, perpendicular line, parallel line, and line
segment, based on the undefined notions of point, line, distance along a
line, and distance around a circular arc.
*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.

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).
*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.

GCO.A.3

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

GCO.A.4

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

GCO.A.5

Given a geometric
figure and a rotation, reflection, or translation, draw the transformed
figure using, e.g., graph paper, tracing paper, or geometry software.
Specify a sequence of transformations that will carry a given figure onto
another.
*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.

B. Understand congruence in terms of
rigid motions.

GCO.B.6

Use
geometric descriptions of rigid motions to transform figures and to predict
the effect of a given rigid motion on a given figure; given two figures,
use the definition of congruence in terms of rigid motions to decide if
they are congruent.
*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.

GCO.B.7

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

GCO.B.8

Explain how the
criteria for triangle congruence (ASA, SAS, and SSS) follow from the
definition of congruence in terms of rigid motions.
*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).

C. Prove and
apply geometric theorems.

GCO.C.9

Prove theorems about lines and angles. Theorems
include: vertical angles are congruent; when a transversal crosses parallel
lines, alternate interior angles are congruent and corresponding angles are
congruent; points on a perpendicular bisector of a line segment are exactly
those equidistant from the segment’s endpoints.
*I
can identify and use properties of vertical angles.
*I
can identify and use properties of parallel lines with transversals,
corresponding angles, and alternate interior and exterior angles.
*I
can identify and use properties of perpendicular bisector.
*I
can identify and use properties of equidistant from endpoint.
*I
can identify and use properties of all angle relationships.
*I
can prove vertical angles are congruent.
*I
can prove corresponding angles are congruent when two parallel lines are
cut by a transversal and converse.
*I
can prove alternate interior angles are congruent when two parallel lines
are cut by a transversal and converse.
*I
can prove points are on a perpendicular bisector of a line segment are
exactly equidistant from the segments endpoint.

GCO.C.10

Prove
theorems about triangles. Theorems include: measures of interior
angles of a triangle sum to 180°; base angles of isosceles triangles are
congruent; the segment joining midpoints of two sides of a triangle is
parallel to the third side and half the length; the medians of a triangle
meet at a point.
*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

GCO.C.11

Prove
theorems about parallelograms. Theorems
include: opposite sides are congruent, opposite angles are congruent, the
diagonals of a parallelogram bisect each other, and conversely, rectangles
are parallelograms with congruent diagonals.
*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.

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 explain the
construction of geometric figures using a variety of tools
and methods.
*I
can apply the definitions, properties and theorems about line segments,
rays and angles to support geometric constructions.
*I
can apply properties and theorems about parallel and perpendicular lines to
support constructions.
*I
can perform geometric constructions including: 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 given line through a point not
on the line, using a variety of tools an methods (compass and straightedge,
string, reflective devices, paper folding, dynamic geometric software,
etc.).

GCO.D.13

Construct an equilateral triangle,
a square, and a regular hexagon inscribed in a circle.
*I
can construct an equilateral triangle, a square and a regular hexagon
inscribed in a circle.

GSRT: Similarity, Right Triangles, and
Trigonometry

B. Prove and apply
theorems involving similarity

GSRT.B.5

Use congruence
and similarity criteria for triangles to solve problems and to prove
relationships in geometric figures.
*I
can recall congruence and similarity criteria for triangles
*I
can use congruency and similarity theorems for triangles to solve problems
*I
can use congruency and similarity theorems for triangles to prove
relationships in geometric figures


Enduring Understandings:
·
Since many geometric figures in the real world are not stationary,
transformations provide a way for us to describe their movement.
·
Proof and congruence are not exclusive to mathematics and the logical
processes with defining principles can be applied in various life
experiences.
·
Definitions establish meanings and remove possible misunderstanding.
Other truths are more complex and difficult to see. It is often possible to
verify complex truths by reasoning from simpler ones by using deductive
reasoning.
·
The geometric relationships that come from proving triangles congruent
may be used to prove relationships between geometric objects.
·
Representation of geometric ideas and relationships allow multiple
approaches to geometric problems and connect geometric interpretations to
other contexts.
·
Communicating mathematically appropriate arguments are central to the
study of mathematics.
·
Transformations, symmetry, and spatial reasoning can be used to
analyze and model mathematical situations.
·
Characteristics, properties, and mathematical arguments about
geometric relationships can be analyzed and developed using logical and
spatial reasoning.

Essential
Questions:
·
What is the significance of symbols and “good definitions” in
geometry?
·
What are the undefined building blocks of geometry and how are they
used?
·
What is nature’s geometry? How can man use nature’s geometry to
improve his environment?
·
How are geometric transformations represented as functional
relationships?
·
How can transformations determine whether figures are congruent?
·
How can special segments of a triangle be used to solve realworld
problems?
·
How does geometry explain or describe the structure of our world?
·
How can points of concurrency be used in realworld situations?
·
How can reasoning be used to establish or refute conjectures?
·
How does my understanding of algebraic principles help me solve
geometric problems?
