The Common Core State Standards Initiative is an education initiative in the United States that details what K-12 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. Wherever there are relevant pages in Math Open Reference, the appropriate text is a link to those pages. It is designed to make it easy for teachers and parents to find useful resources to help teach and coach students.

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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.
- See also: reflection of a point, reflection of a line segment.

- 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.
- See also: reflection of a point, reflection of a line segment.

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.
- Explain and use the relationship between the
sine and
cosine of
complementary angles.
- See also: complementary angle formulas.

- 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.

8.1 Derive and use the
trigonometric ratios for special
right triangles (
30°- 60°- 90° and
45°- 45°- 90°) CA

- See also: equliateral triangle, square.

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.
- See also: sine.

- (+) 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 non-right triangles (e.g., surveying problems, resultant forces).

CirclesG C

Understand and apply theorems about circles.

- Prove that all
circles are
similar.
- See also: concentric circles.

- 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.

- See also: tangent line.

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.
- See also: basic equation of a circle, distance formula.

- 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

- See also: Quadratic equation - a graphical explorer.

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.
- See also: dividing a line segment into n equal parts.

- 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.
- See also: other solid figures.

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 objects.
- See also: cylinder generated by rotation.

- 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
- See also: similar polygons, volume formulas.

- 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 real-world and mathematical problems. CA

- See also: converse of triangle inequality theorem.

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).

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

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