G-CO: Congruence

G-CO.A: Experiment with transformations in the plane.

G-CO.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.

Chords and Arcs
Constructing Parallel and Perpendicular Lines
Parallel, Intersecting, and Skew Lines

G-CO.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).

Dilations
Reflections
Rotations, Reflections, and Translations
Translations

G-CO.A.3: Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

Reflections
Rotations, Reflections, and Translations

G-CO.A.4: Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

Reflections
Rotations, Reflections, and Translations
Translations

G-CO.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.

Reflections
Rotations, Reflections, and Translations
Translations

G-CO.B: Understand congruence in terms of rigid motions.

G-CO.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.

Reflections
Rotations, Reflections, and Translations
Translations

G-CO.C: Prove geometric theorems and, when appropriate, the converse of theorems.

G-CO.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, and conversely prove lines are parallel; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.

Constructing Parallel and Perpendicular Lines
Investigating Angle Theorems
Segment and Angle Bisectors

G-CO.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, and conversely prove a triangle is isosceles; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; and the medians of a triangle meet at a point.

Concurrent Lines, Medians, and Altitudes
Congruence in Right Triangles
Isosceles and Equilateral Triangles
Proving Triangles Congruent
Triangle Angle Sum
Triangle Inequalities

G-CO.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.

G-CO.C.11.a: Prove theorems about polygons. Theorems include the measures of interior and exterior angles. Apply properties of polygons to the solutions of mathematical and contextual problems.

Parallelogram Conditions
Polygon Angle Sum
Special Parallelograms

G-CO.D: Make geometric constructions.

G-CO.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.). Constructions include: 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.

Constructing Congruent Segments and Angles
Constructing Parallel and Perpendicular Lines
Segment and Angle Bisectors

G-SRT: Similarity, Right Triangles, and Trigonometry

G-SRT.A: Understand similarity in terms of similarity transformations.

G-SRT.A.1: Verify experimentally the properties of dilations given by a center and a scale factor:

G-SRT.A.1.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.

Dilations

G-SRT.A.1.b: The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

Dilations

G-SRT.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.

Dilations
Similar Figures

G-SRT.A.3: Use the properties of similarity transformations to establish the Angle-Angle (AA) criterion for two triangles to be similar.

Proving Triangles Congruent
Similar Figures

G-SRT.B: Prove theorems involving similarity.

G-SRT.B.4: 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.

Congruence in Right Triangles
Proving Triangles Congruent
Pythagorean Theorem
Similarity in Right Triangles

G-SRT.B.5: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

Chords and Arcs
Perimeters and Areas of Similar Figures
Pythagorean Theorem
Similar Figures
Similarity in Right Triangles

G-SRT.C: Define trigonometric ratios and solve problems involving right triangles.

G-SRT.C.6: 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.

Sine, Cosine, and Tangent Ratios

G-SRT.C.7: Explain and use the relationship between the sine and cosine of complementary angles.

Cosine Function
Sine Function

G-SRT.C.8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

Distance Formula
Pythagorean Theorem
Pythagorean Theorem with a Geoboard
Sine, Cosine, and Tangent Ratios

G-C: Circles

G-C.A: Understand and apply theorems about circles.

G-C.A.2: 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.

Chords and Arcs
Inscribed Angles

G-C.A.3: Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral and other polygons inscribed in a circle.

Concurrent Lines, Medians, and Altitudes

G-C.B: Find arc lengths and areas of sectors of circles.

G-C.B.5: 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.

Radians

G-GPE: Expressing Geometric Properties with Equations

G-GPE.A: Translate between the geometric description and the equation for a conic section.

G-GPE.A.1: 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.

Circles

G-GPE.A.2: Derive the equation of a parabola given a focus and directrix.

Parabolas

G-GPE.A.3: 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.

Ellipses
Hyperbolas

G-GPE.A.3.a: Use equations and graphs of conic sections to model real-world problems.

Circles

G-GPE.B: Use coordinates to prove simple geometric theorems algebraically.

G-GPE.B.4: Use coordinates to prove simple geometric theorems algebraically including the distance formula and its relationship to the Pythagorean Theorem.

Distance Formula

G-GPE.B.7: Use coordinates to compute perimeters of polygons and areas of triangles and rectangles (e.g., using the distance formula).

Distance Formula

G-GMD: Geometric Measurement and Dimension

G-GMD.A: Explain volume formulas and use them to solve problems.

G-GMD.A.1: 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.

Circumference and Area of Circles
Prisms and Cylinders
Pyramids and Cones

G-GMD.A.2: Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.

Prisms and Cylinders
Pyramids and Cones

G-GMD.A.3: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

Prisms and Cylinders
Pyramids and Cones

G-GMD.B: Visualize relationships between two-dimensional and three-dimensional objects.

G-GMD.B.4: Identify the shapes of two-dimensional cross sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

Prisms and Cylinders
Pyramids and Cones