COMMON CORE STATE STANDARDS
|Experiment with transformations in the
||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. Trapezoid is defined as "A quadrilateral with at
least one pair of parallel sides."
||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
||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
||Explain how the criteria for triangle congruence
(ASA, SAS, and SSS) follow from the definition of congruence in terms of
|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; complementary and supplementary angles.
||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; an exterior
angle of a triangle is equal to the sum of the two non-adjacent interior
angles of the triangle..
||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;
rhombuses are parallelograms with perpendicular 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; constructing a line parallel to a given line
through a point not on the line; constructing the median of a triangle
and constructing an isosceles triangle with given lengths.
||Construct an equilateral triangle, a square, and a
regular hexagon inscribed in a circle.
|Similarity, Right Triangles, &
|Understand similarity in terms of
||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; the length of the altitude drawn from the vertex of
the right angle of a right triangle to its hypotenuse is the geometric
mean between the lengths of the two segments of the hypotenuse.
||Use congruence and similarity criteria for triangles
to solve problems and to prove relationships in geometric figures.
ASA, SAS, SSS, AAS, and Hypotenuse-Leg theorem are valid criteria for
triangle congruence. AA, SAS, and SSS are valid criteria for
|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.
||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.
|Understand and apply theorems about
||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; angles involving tangents and
secants; inscribed angles on a diameter are right angles; the radius of
a circle is perpendicular to the tangent where the radius intersects the
||Construct the inscribed and circumscribed circles of
a triangle, and prove properties of angles for a quadrilateral inscribed
in a circle.
|Find arc lengths and areas of sectors of
||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.
Radian introduced only as unit of measure.
|Expressing Geometric Properties with
|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.
|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); include distance formula;
relate to Pythagorean theorem.
||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 & Dimension
|Explain volume formulas and use them to
||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.
||Use volume formulas for cylinders, pyramids, cones,
and spheres to solve problems.
|Visualize relationships between
two-dimensional and three-dimensional objects
||Identify the shapes of two-dimensional
cross-sections of three-dimensional objects, and identify
three-dimensional objects generated by rotations of two-dimensional
|Modeling with Geometry
|Apply geometric concepts in modeling
||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
||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