
Math Common Core  Alignment with Math Open Reference
The Common Core State Standards Initiative is an education initiative in the United States that details what K12
students should know in English and Math at the end of each grade year,
and seeks to establish conformity in education standards across the US.
The document below is a reproduction of the relevant parts of the common core geometry standards.
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Congruence G CO
Experiment with transformations in the plane.
 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.
 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.).
 Given a
rectangle,
parallelogram,
trapezoid, or
regular polygon, describe the rotations and
reflections that carry it onto itself.
 Develop definitions of
rotations,
reflections, and
translations in terms of
angles,
circles,
perpendicular lines,
parallel lines,
and
line segments.
 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.
Understand congruence in terms of rigid motions.
 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.
 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.
 Explain how the
criteria for triangle congruence (
ASA,
SAS, and
SSS) follow from the
definition of congruence in terms of rigid motions.
Prove geometric theorems.
 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.
 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.
 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.
Make geometric constructions.
 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.
 Construct an
equilateral triangle, a
square, and a
regular hexagon inscribed in a circle.
Similarity, Right Triangles, and Trigonometry G SRT
Understand similarity in terms of similarity transformations.
 Verify experimentally the properties of
dilations given by a center and a scale factor:
 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.
 The
dilation of a
line segment is longer or shorter in the
ratio given by the scale factor.
 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.
 Use the properties of
similarity transformations to establish the
AA criterion for
two triangles to be similar.
Prove theorems involving similarity.
 Prove 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.
 Use
congruence and
similarity criteria for
triangles to solve problems and to prove relationships in geometric figures.
Define trigonometric ratios and solve problems involving right triangles.
 Understand that by
similarity, side ratios in
right triangles are properties of the
angles in the triangle, leading to definitions of
trigonometric ratios for
acute angles.
 See also: introduction to trigonometry,
trigonometry overview,
SOH CAH TOA,
inverse trigonometric functions,
sine,
arcsine,
cosine,
arccosine,
tangent,
arctangent,
cosecant,
secant,
cotangent.
 Explain and use the relationship between the
sine and
cosine of
complementary angles.
 Use
trigonometric ratios and the
Pythagorean Theorem to
solve right triangles in
applied problems.
 See also: trigonometry overview,
SOH CAH TOA,
inverse trigonometric functions,
sine,
arcsine,
cosine,
arccosine,
tangent,
arctangent,
cosecant,
secant,
cotangent.
Apply trigonometry to general triangles.
 (+) Derive the formula
A = 1/2 ab sin(C) for the
area of a triangle by drawing an auxiliary line from a
vertex
perpendicular to the opposite side.
 (+) Prove the
Law of Sines and
Law of Cosines
and use them to
solve problems.
 (+) 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).
CirclesG C
Understand and apply theorems about circles.
 Prove that all
circles are
similar.
 Identify and describe relationships among
inscribed angles,
radii, and
chords. Include the relationship between
central,
inscribed, and circumscribed angles;
inscribed angles on a diameter are right angles; the
radius of a circle is perpendicular to the tangent where the radius intersects the circle.
 Construct the
inscribed circle and
circumscribed circle of a triangle,
and prove
properties of angles for a
quadrilateral inscribed in a circle.
 (+) Construct a tangent line from a point
outside a given circle to the circle.
Find arc lengths and areas of sectors of circles.
 Derive using similarity 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; derive the formula for the
area of a sector.
[Convert between degrees and radians. CA]
Expressing Geometric Properties with EquationsG GPE
Translate between the geometric description and the equation for a conic section.
 Derive the
equation of a circle of given center and radius using the
Pythagorean Theorem;
complete the square to find the
center and
radius of a circle given by an equation.
 Derive the equation of a
parabola given a focus and directrix.
 (+) Derive the
equations of ellipses and hyperbolas given the foci,
using the fact that the sum or difference of
distances from the foci is constant.
3.1 Given a quadratic equation of the form
ax ^{2} + by ^{2} + cx + dy + e = 0,
use the method for completing the square to put the equation into standard form;
identify whether the graph of the equation is a
circle,
ellipse, parabola, or hyperbola and
graph the equation. CA
Use coordinates to prove simple geometric theorems algebraically.
 Use
coordinates to prove simple geometric theorems algebraically.
For example, prove or disprove that a figure defined by four given points in the
coordinate plane is a
rectangle;
prove or disprove that the point (1, 3) lies on the
circle centered at the
origin and containing the point (0, 2).
 Prove the
slope criteria for
parallel and
perpendicular lines and use them to solve geometric problems (e.g., find the
equation of a line
parallel or
perpendicular to a given line that passes through a given point).
 Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
 Use
coordinates to compute
perimeters of polygons and areas of triangles and rectangles, e.g., using the
distance formula.
Geometric Measurement and DimensionG GMD
Explain volume formulas and use them to solve problems.
 Give an informal argument for the formulas for the
circumference of a circle,
area of a circle,
volume of a cylinder,
pyramid, and
cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments.
 (+) Give an informal argument using Cavalieri's principle for the formulas for the
volume of a sphere and
other solid figures.
 Use volume formulas for
cylinders,
pyramids,
cones, and
spheres to solve problems.
Visualize relationships between twodimensional and threedimensional objects.
 Identify the shapes of twodimensional crosssections of threedimensional objects,
and identify threedimensional objects generated by rotations of twodimensional objects.
 Know that the effect of a
scale factor k greater than zero on
length,
area, and volume is to multiply
each by k, k2 , and k3 ,respectively; determine length, area and volume measures using scale factors. CA
 Verify experimentally that in a triangle,
angles opposite longer sides are larger,
sides opposite larger angles are longer, and the
sum of any two side lengths is greater than the remaining side length;
apply these relationships to solve realworld and mathematical problems. CA
Modeling with GeometryG MG
Apply geometric concepts in modeling situations.
 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a
cylinder.
 Apply concepts of density based on
area and
volume in modeling situations (e.g., persons per square mile, BTUs per
cubic foot).
 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).
Acknowledgements
This Common Core alignment document was created by Patrick Beal, an educational consultant from southern California.
He can be contacted at beal.math@gmail.com
(C) 2011 Copyright Math Open Reference. All rights reserved

